 So I have the pleasure of being able to introduce the Neil today. And for those of you that have not attended any of these, let me give you a little bit of history. So this celebration of faculty careers came from two actions of the strategic plan for the college of engineering. The faculty of 2020, with its focus on professional development at all stages of our career, and the alignment of criteria and processes for hiring and promotion and tenure with the evolving scope of the College of Engineering and its leadership values. A desire was expressed by someone for a review post-promotion to the rank all the way to full professor that would feature the accomplishments of the faculty and provide also an opportunity to plan for the next phase of a faculty member's career. Full professors who are at least seven years past promotion present colloquia on their achievements and their plans to their peers. And this is followed by a planning discussion with their deans and their heads and other senior individuals. This was piloted successfully in the spring of 2013, and it had full implementation in the 2012-14 academic year. So you all know Neil, but just a little bit of background for the video. Neil Pajaj is the Alpha P Davidson professor and William E. and Florence E. Perry head of the School of Mechanical Engineering. He joined the Purdue faculty in January 1981 after completing his PhD studies in mechanics from the University of Minnesota. He obtained, respectively, bachelor's and master's degrees in mechanical engineering from the IIT at Caragpur and the IIT at Comfort. Dr. Pajaj's research and teaching interests are quite broad. They're in the area of linear and nonlinear systems, analytical dynamics and modeling of multi-body systems, stability of elastic systems, bifurcations and chaos in mechanical systems, nonlinear stability in fluid and fluid structure interactions, dynamics of sea occupant systems, mowing of disco elastic properties of foam, and design of nonlinear resonant mims. And I think that's captured up here in three lines. So we'll hear the rest in terms of the details from Neil. Thank you, Melba. Dean Crawford, it's really an honor to be able to talk to you guys as part of this celebration of faculty careers. As you saw, I'm surprised it's April Fool's Day, so hopefully it'll be all fine. So, existence of a faculty member for so long, I think I'm older than some of you in the audience. So that means that I have received significant help and support from everybody, from most. And so that's the first thing I want to say. I'm thankful to all the collaborators as well as the rest of the faculty, not only in mechanical engineering but in college of engineering and across campus for allowing me the existence and to the point that I was willing to apply for the headship of the school. But today I'll mostly talk about, so this is some outline of the talk. So I'll talk a little bit about the background of Anil Bajaj and some introduction and scope of this presentation, talk a little bit about some initial research in dynamical systems, and then more recent research in the last few years, and then say a few words about my service to the school or administration and so forth. Now, afterthought here is really what I realized was I had not included in all this. So I thought at least I should put it out and at least try to complete and give credit to everybody. And I'm sure even after this afterthought, I've missed out lots of people. Okay, so the background as Melba already talked about, so I did my bachelor's degree in 1973, five-year degree program, not four-year degree program, from Indian Institute of Technology, Kharagpur, and this is how the main building of IIT Kharagpur looks like. And then I did, and in all of these things there's lots of stories, and I'm sure my kids have heard some of those stories, but not everybody. And so depending on how I see you guys smiling or not, I'll sometimes say some of those stories. Then I went to IIT Kanpur for my master's degree, and that's how it looks like. That's actually, this is library, which was built, well, IIT Kanpur was one of the IITs, and this was the IIT which was built with U.S. help. In fact, the librarian of IIT Kanpur library, first librarian was the librarian of the Maths Science Library here, Mr. Richard Funkhauser. And so there is, and in fact, we had a faculty member in ME, Val Buckdoldt in the design area, and he had spent first three years in the startup of IIT Kanpur. So I had wonderful experience there. I did my master's degree there, and then I went to Minnesota, and this is how actually, unfortunately, this is the new aerospace engine and this thing building, not the old one, which I remember. That's the way it looked in the 70s or now. Which these looked at that time also. Yes, that has not changed much. So here is my graduating class, and I did not have that picture. It's only last few years that I joined a IIT KGP Google group. Yes. Are you wearing a tie? No, hell no. Here I am. I'm sure that would have been a quiz. Recognize who is Bajaj here. He's not wearing a tie. No tie. Yes. In fact, I don't even have a jacket there. So yes, true to the spirit that has remained throughout my life. So in fact, it's interesting that a colleague, a friend of ours, he put down these names here by trying to memory, and he says memory fails, memory fails, and so on. And luckily, I could recognize all the professors, and I can tell you the names. It's a very small class. Fortunately, unfortunately, that was a rare year at IIT in IIT's history, and the 1968 batch entering to all five IIT's was the smallest ever. For whatever reasons that took place, the previous batches were usually of the order of around 300 students going to each IIT, whereas this batch was around 150 students, and so we had only 28 students in our, which meant that every professor remembered our name, which meant that other implications from there. So let me talk a little bit about more introduction. So at IIT Kanpur, I decided to do research. Now, I had too many interests, which is still true. And so I was interested in machine tool dynamics, machine chatter, and so on. I was interested in control theory. I was interested in optimization theory, and at that time, what happened, the professor who was very much interested in doing finite element and optimization, he was in so much demand that practically the whole class, empty class, wanted to work with him. And you all know that person, many of you know that person, SS Rao. So I also sat in front of his office waiting for my turn to go and ask whether he has a project I can work. Well, in the meantime, the rest of the faculty complained vociferously to the head of the department. We are not getting any grad students. So the rule was made by the, they called postgraduate faculty committee that unless all faculty members get at least one or two students, no faculty member could take additional students. So I said, okay, I will be the first one to give up. And I worked with, I said, okay, who don't have a student? I said, I'll work with them. And I worked on linear stability of planar jets. So that's completely different than, I did take a few courses in fluid mechanics and so on, but all right. So what was this linear stability of jet flows? Planar jet is basically, I'm sure all the Paul Sawicka can explain much more than I can. But basically, if you take a infinitely long slit, but the slit size is infinite small, okay. And a fluid is injected from there out and fluid is coming out into the same medium, which is stationary, right. So water coming out into water, air coming out into air and so forth. And so what happens is because the slit size is infinite small, everything is really defined in terms of the momentum of the fluid that is being injected out, okay. Of course, what happens as this jet flows out, it is going to start and training fluid from the sides. This is into an infinite, semi-infinite fluid. And so the jet, whatever way you define the area of the jet or volume encompassed with the jet, that slowly starts growing in some manner in a conical fashion. So you not only have a axial flow, you also have a transverse flow. And this axial flow is not a constant velocity independent of X, but hopefully it's slowly growing, not fast growing, right. Okay. So you want to study its stability. So that's a very classical problem. And so parallel flow model is not fully valid because it slowly is expanding. And so the stability analysis, one of the classic ones was by Sato in 1960. And the classical method for parallel flow stability analysis is something called, you introduce perturbations in Navier-Stokes equations and make the assumptions of planar flows and all that stuff. And what you get is something called Orr-Sommerfeld boundary value problem. And it is a tough boundary value problem to solve. It is singular. And in those days, they were constructing numerical methods for solving those, okay. So there are NASA codes that were available for a few of them. But because the flow is growing, Professor Joseph is one student. He had a PID thesis and said, so because of this slow growth, the Orr-Sommerfeld model is not the right model for this. And we need to modify that equation and do something else about it. So there was a modified Orr-Sommerfeld equation which appeared in applied mechanics reviews. And just around the time I was doing my master's degree, I had started my master's at 74 or so. I think this date may be wrong. And so this is the modified Orr-Sommerfeld equation. And this is a linear operator. V is the perturbation to the axial flow. And so V can be represented in the form of something which depends only on y and a temporal and a spatial wave number. Okay. I'm not going to go through the theory, but just the ultimate result, how things look like. So the results in linear stability theory usually are presented in terms of what's the basic parameter. That's Reynolds number. As I increase the Reynolds number, what happens to the flow? So essentially what happens to the flow is essentially is this profile going to remain stable? If I give a small disturbance, it is going to remain as it is or these disturbances will start growing and you will start forming vortices and so forth. Okay. And here it is, this is the, as a function of kr, so the spatial wave number. And these are for different values of the imaginary parts of the wave number. And so they represent really the growth rate or decay rate in spatial direction. And so these are parameterized by it and then there is the frequency of oscillation along this curve. So essentially neutral curve means that Ki is equal to zero. Okay. And so this is for the parallel flow theory and this is for the modified parallel flow theory. So what's that? A theoretician, how do I know which is correct? Luckily, Sato had done experiments and he had seen in the experiments that the Reynolds number at which the jet became unstable was around 10, whereas the parallel flow theory predicts around 4. Okay. And for whatever fortunative coincidence, this modified theory predicts around 11.4. So that's just one data point to say this may be certainly a better model. Okay. So that was IIT Kanpur. Okay. So there are again stories in life. So whatever they say, fork, right, on the road. So one of the professors at IIT Kanpur said, stick around here. You have done enough work. If you do one more or two more years of work, you will get a PhD. And we like you, you are good enough and we can promise you, we'll give you a lecture or chip position. For whatever coincident that year, they had two assistant professors. One was PID from British Columbia and one was PID from Stanford. Okay. And they were as hard as assistant professors. So I said, well, these two gentlemen, you have hard as assistant professors and you are saying to me, if I finish my PhD here, I'll be hard as a lecturer. If you promise, you will hire me as assistant professor. I do PhD here. Otherwise, I'll go outside, do PhD there and come back and then you give me assistant professor position. So that was the packed with understanding with them. Okay. You can go. And so that was the decision to come to USA. Okay. With all the intention that all, and I did apply, unfortunately they still gave me a lecturer's position while Purdue gave me assistant professor position. So that's where things happen. All right. Okay. So as you saw, I went to fluid stability and then I saw all kinds of the books by Dan Joseph and so on. So I said, okay, it'll be great to study at Minnesota with Dan Joseph. And I got a teaching assistantship to do PhD. When I get there, Dan is on sabbatical. And all his students, a few of them were Indian origin. In fact, from IIT Kharagpur, couple of them were, they said, well, Dan has stopped working in stability theory. He only works in viscoelastic fluids now. Okay. So I said, what do I do now? Okay. And Dan was going to come back. This was September of 1976 and I was, he was not going to be there for six more months and so on. So I started taking classes in continuum mechanics, dynamical systems and so forth. And then the professor who was teaching dynamical systems at the end of the quarter system, he said, well, I have a project and do you want to work with me? No commitments for PhD. Okay. So I went to Guinea Pig to help me out or my PAD student to finish some work and you seem to have understood whatever I taught in class in dynamical systems. So I said, fine. But that's what then remained for four and a half years after that. Even after Dan Joseph came back. So I joined PR Sathna's dynamical systems group. Professor Sathna's group only always consisted of one student. Okay. What were the visa issues? Pardon? What were the visa issues? Oh, the visa issues. Okay. So that's another story. All right. So India was in good books of United States in those days, which meant that a large fraction of the people, students who went to get student visa were denied visa. Okay. So whether that was because the Indian government was, had requested the U.S. government to do this, which is quite possible. Okay. Or it was because of the relations because Mrs. Gandhi was the prime minister and so forth for whatever reason. So a few students from IIT Kanpur, like I was supposed to go for visa on Monday, a few students from IIT Kanpur came back the previous week and they were all crying. They had all been rejected. Their visa applications had been rejected. So they went to the head of the department and said, do something about it. And at that time, the science advisor, science and technology advisor to the U.S. ambassador was actually a professor from University of Massachusetts who had been one of the faculty members at IIT Kanpur. So these people were given letters by my head of the department to say something about, you know, good guy, why don't you help them get the visa and so forth. Okay. So that's what they were doing. And they were going back to Delhi and see if they can get visa. So that's how the story went. So I was also given a letter. I never opened that letter. I said, I'll see if I need to use it, then I'll worry about it. But I was, they had called me and said, you are going there. I know that. Okay. So here's a letter. You can take it and take it directly to the attached science officer and he will help you. Okay. I didn't go there. But luckily on that day, it's true out of 30, only two people were issued visas to come to U.S., myself and Mr. Bandopadhyay. And he went to Amherst. Okay. That's all. Everybody else's visa was, application was rejected. So I don't know what is the reason. But that's, you know, there are some other. So letters didn't help? Pardon? So letters didn't help? No. They didn't help. Okay. So again, as I said, Professor Satna, his group would have one or two students. Usually one, he would have usually an NSF grant or an Air Force grant. He will hire one student. He will work with the student like hell. Right? So like hell means like 6 o'clock, 6 p.m. in the evening, he'll be working with the student. And they will decide, yeah, this is a good idea. Try something here and so on. And 8 o'clock in the morning, next day, he will be at your door and say, did you try it? Did it work or not? And so on. Okay. So it was tremendous pressure that the output was needed. Okay. Well, it was not because he wanted to pressure you, but he was excited about the outcomes. Okay. So the thing was that nonlinear vibrations, I still call it nonlinear vibrations, though most of my colleagues now call it nonlinear dynamics. I tried to distinguish between the two. The local bifurcation theory, that was gaining fashion. But the applications in mechanical systems is very much in infancy. Okay. And so, you know, nonlinear structures, elastic stability, a very important area, concept called follower forces, that was gaining significant amount of notoriety. There was a lot of effort in that direction because follower forces, what happens is static stability criteria and dynamic stability criteria give different results. And that used to be called a paradox. Okay. And so, and then there were a few classics appearing just then. One is George Herman's very classic NASA technical report where he talked about elastic stability, follower forces and so on. Thamson and Hunt's book on the stability of elastic structures and Marston and McCracken's book on half bifurcation. Those things appeared at that time. Okay. So these are the classics. So this is around, you know, early 1970s, this is 73 and this is 1976. Okay. And luckily or unlikely, my advisor got the first few copies of this book to review for Siam review. Okay. So which meant that I had to read the book or help him read the book. Okay. All right. So I had to dive into dynamical systems or weekly nonlinear systems, learned asymptotic, averaging theory and so on. And so the idea was to marry nonlinear vibrations with nonlinear dynamics. Okay. And again, this is a kind of a conversation in a belt way. Some of you will appreciate. Most of you will say who the hell cares. Okay. And that's perfectly fine. So the first things I started doing was self-excited oscillations of structures with internal flows and Herman George Herman in Stanford had a PhD student who had just finished doing a problem on that. And then this was my PhD thesis, Bifurcations in Systems with Rotational Symmetry. Okay. That was the first time it was done in dynamics and for first five years that was the most cited paper in that area, but after that mathematicians took over they never cite me on that topic. Okay. They only cite each other. Then came the faculty opportunity at Purdue. Okay. I interviewed in June 1980. I think a week after Professor Krauskiel interviewed, there were two faculty openings here in the mechanics area in dynamics vibration and so on. And late Professor Joe Jenin, he was the chair of the mechanics area and he recruited both of us. Okay. Again, as I said, everything has a story behind it. Right? So the story behind it is when I applied to Purdue, Professor Jenin called me and said, are you a permanent resident? I said, no, I have a student visa. So I said, well, our head says and the head at that time was Professor Arthur LaFaber that the head says no international students or our international students and scholars program says we don't want to struggle with trying to go through the visa process and so on. Okay. So if I don't call you, understand that not that you are, we like you, everything, we would like to recruit you, but we can't do this. Right? Well, luckily a month later, he called again. He said, well, finally our head has agreed to interview you. So that's what happened. And the day I came here, they were only, actually there was only one faculty member who came to my seminar. There were a total of four people in the seminar, Arthur LaFaber, Bob Fox, Joe Jenin and Ray Sipra. Okay. So thank them that I'm here. Or whatever. Right? Okay. So, I joined in January 1981, as Melba said, and I was promoted in April 1985 and then again in April 91 to associate to full and then the school was kind enough to appoint me as Alpha PJ Mason professor in December 2009. Okay. So let's talk about a few of the dynamical systems. I can see half of, half hour is already gone in the stories. And I'm only on slide 11 and I have 54 slides. Okay. So, excuse me. Okay. So let me talk a little bit about the self-excited oscillations in structures or with internal flows. This was the first application of half bifurcation theory. This was the first application of bifurcation in systems with symmetry. And then bifurcation in systems with broken symmetry. Okay. All right. So here, and the simple problem is the garden hose problem. Okay. So, you have a garden hose. You hold it at one end. You suddenly turn on the faucet and if this flow rate is sufficiently high, it starts doing wild stuff. Okay. So can I predict it? And this is what I will call in the simplest form a limit cycle oscillation. And how does this transition from straight flow to limit cycle oscillation take place? Okay. So that's the garden hose problem or fire hose problem. You've seen the firefighters have to really hold on to the front end of the pipe really well. Otherwise it will start going haywire. Okay. Now the key thing is this is a non-self-adjoint boundary value problem and it is gyroscopic system. Okay. And so it has all the elements that mean that the static stability theory is not applicable here. It has to be studied in the context of dynamic stability theory. Okay. All right. So here, now of course this is a system with infinite Eigen values. Okay. Infinite complex Eigen values. Stability would mean all Eigen values must remain in the left half plane and as the Eigen values cross into the right half plane then it will be an unstable system. And then my interest is non-linear. So this means I need to continue and see what happens when the parameters are outside or beyond the critical value for linear stability. Okay. So here is a what they call argon diagram or a root locus or whatever. So this is the imaginary axis all Eigen values are imaginary. Just at zero flow rate they all start on the imaginary axis infinity of them. So that's one of the issues. Right. How do I know that a 99th Eigen value is not doing something funny. So proving those things is relatively tricky business. Okay. All right. So we start with this. So you can do computations but only for small order Eigen values. All right. So you start here as you increase the flow rate you can see that the first Eigen value the third Eigen value they're all going into the left half plane whereas the second Eigen value first goes into the left half plane and then it comes out at this flow rate. Okay. So the oscillation of this tube the tube becomes unstable in the second mode. Now this is really not second mode of the linear undamped vibratory system it is the second mode of the gyroscopic linear gyroscopic system. So it's not beam mode it's a complex combination of many modes in terms of beam modes. Okay. So it's unstable in second mode purely many Eigen values and so if the motion of the tube is in a plane now unfortunately we could never buy tubes which were straight enough that experimentally they would do planar vibration. Okay. What happens is these tubes get manufactured and then if there are long tubes polyacetylene or something they come in roles and it's like a dog's tail you can never straighten it. Okay. And because you can never straighten it you can never start with a straight tube. Okay. So already there is a perturbation built into it and so forth. So theory verification from experiments is somewhat complicated. Okay. All right. So hopefully it goes into half fabrication to limit cycle. That's what the expectation from the theory is. So the problem has rich two parameter problem. There are two interesting parameters. One is the mass ratio which is in terms of the mass of the fluid versus mass of the tube material solid and the other one is actually we discovered something called length ratio and that had to do something with how long is the tube relative to some Reynolds diameter or something of the okay. So if the tube is very long then we had one kind of behavior. At least that's what we predicted and if the tube is short the behavior was a different type. Okay. And so we were able to do experiments. There's another story here and that story is everything in my room upstairs on third floor has been packed up and put in the attic. Unfortunately it had a 8 millimeter tape which documented this experiment. So I tried to search for that tape but so far I have not been able to find. Okay. Where it is in my stuff. Okay. Thank you very much. Yes. Yes. I had converted that tape into a VCR this thing video. Okay. But yes. So both of the things are somewhere in the attic. Okay. So what the outcome is. The outcome is okay. So there is a super critical half fabrication which essentially means you have stable oscillation in the plane as the flow rate exceeds some critical value the amplitude starts growing. The amplitude of this oscillation the shape of the tube is in the second mode and the amplitude of oscillation is proportional to the half power of the difference between critical flow rate and where you are. So that's what we were able to prove theoretically and so it looks like this. Essentially. The bifurcating solution this is only representing amplitude and if you put it in some sort of a reduced order model plane then you see a limit cycle oscillation. Now in this case this is called super critical and the bifurcating solution is stable. Okay. There is another situation where the bifurcation is subcritical so that there is stable zero solution there is an unstable solution and then there is a turning point and there is a stable solution. So you have in this range the linear flow is still stable linear this thing is still stable but there is an unstable limit cycle and there is a stable limit cycle this is called subcritical bifurcation you need to go to higher order terms in your bar length but it turned out that for small tubes this is what happens and for long tubes this is what happens so theory predicted that and we were able to confirm that experimentally. So this is what it says so this is the mass flow rate the mass parameter this is the length parameter and these are the three regions in which things take place and this is the super so depending on where you are you can see different behaviors these circles here represent where we did the experiments and they worked fine. Now as I said we never can find a straight tube and we can never point a tube which is relatively not circularly symmetric so actually when we do experiment it may do this or it may start doing this orbital motion and so that was if the system is symmetric that's what happens now another discovery it's really not always that this happens it depends on again the mass ratio beta square so there is an interval here you can see so you have two possibilities you have bifurcation bifurcation to this is what is called standing wave and this is called rotary wave or traveling wave solution there is no difference in the system it's the same physical system with rotational symmetry so it turned out as a function of beta there are these three intervals and in these intervals there is an exchange in which which rather the traveling wave is stable or the standing wave is stable that takes place now I have no physical explanation as to what differentiates one region versus other why suddenly a standing wave becomes unstable and a traveling wave becomes stable it's just result of computations and luckily experiments also verified that now since I can never find a tube which is completely circular we introduce intentional asymmetry intentional asymmetry so for example here is a tube with nearly circular section and we cut out flats on it which meant that in the two directions the moment of inertia got changed which allowed that it's softer in the direction which would force it to do planar motions so we did it for a tube here for which the circular the traveling wave was stable and as soon as we introduced this perturbation now you see that actually standing wave was stable and then it went underwent a little bit of complex bifurcation process before it changed into traveling wave so again this was verified by experiments so I was quite proud of all this that theory works in experiments okay there are hundreds of papers on this subject afterwards lots of experimental work and so on and process of paeduses and frank moan were two of the people pioneers in this okay now let me talk a little bit about resonance structures and modal interactions so it's a relatively busy this thing but one of the things in nonlinear dynamical systems is you can always pick one problem at a time but that's not fun right you are always looking for unifying principles what can be a unifying principle okay so for example here there is a spherical pendulum right there is a spherical pendulum it has a what is called spherical is symmetry of rotation it looks the same whichever coordinate system I choose similarly if I take a string if I allow if the string is relatively circular section it can stay do this vibration it can do this vibration or it could do this and it is symmetric with the axis right similarly I can take instead of string I could take a beam what's the difference between a string and a beam a string is second order differential equation in space beam is a fourth order difference so it's hyperbolic versus a what do you call that too far away from teaching no it's not elliptical no I'm sure yeah no no no so these are not these are not the classical second order operators this is a fourth order so this is the okay so you need to beam equation is that kind of thing okay so in any case the point is all these systems have a symmetry so the question is do all these systems represent the dynamics in a circular way yes it may not be same parameter values that depends on the physics of the problem okay so most of the time in my research the idea was always trying to look for these underlying principles of symmetry and whether they shed light on the behavior of the system alright so so these are very classical problems in mechanics and so I started with string problem and take the spatial discretization my always theme was luckily that's or unluckily that still holds true I can't even understand 2 degree of freedom systems forget about PDEs okay so if I can work with low order dynamical systems and can explain the behavior I think I'm relatively satisfied with it okay so this is a you know expansion in spatial and displacement why this is allowing both for transfers in the two directions okay this vibration or this vibration why both directions so the key thing here is let's say if I excite the string apply a transfers force like this it vibrates like this in some situations it starts vibrating like this so it has standing motion transfer standing motion but in some cases it goes into rotary motion so under what situation it goes into rotary motion and this rotary motion arises due to some instability and that's really so what we say is there is a mode in this direction there is a mode in this direction and under what conditions there is a strong enough coupling between these two modes that it will lead to this behavior okay and most of my life dynamical system problems that I have looked at they are all because of this modal coupling of this different types so this is a two mode here is a two mode model okay z1 and z2 vector and z1 is this motion z2 is this motion there is a harmonic excitation only in this direction okay there is nonlinearity because otherwise there is no fun so that's one thing secondly it's really the nonlinearity which calls this coupling so you can see here z2 times z this is a cubic nonlinearity stretching of the string so we study these so z1 is the amplitude of in plane motion z2 is the amplitude of out of plane motion beta is the frequency of excitation we are exciting it near resonance frequency and alpha is damping so these alpha and beta are the two parameters which play all the games okay so we study it using averaging asymptotic method bifurcation theory numerical bifurcation analysis something called auto and study roots to chaos and the system goes into chaotic behavior right so here is a string behavior so transverse vibration in the plane this looks like a duffing equation so you can see for small enough damping there are these stable periodic motion stable periodic motion unstable periodic motion if we increase if the increase the forcing amplitude or decrease the damping so here is damping decreased now there is this frequency is beta 1 and beta 5 between these the planar motion becomes unstable and it gives rise to this whirling motion so there is a a1 but now you see an a2 is between beta 1 and beta 5 so the combination of this and this will give rise to whirling now this whirling motion give rise to really complicated dynamics so here is for damping 0.513 the same thing as increasing amplitude of excitation so so this is my some amplitude and I'm starting to see now this is my beta 1 and beta 2 and now the this motion starts undergoing period doubling bifurcations so this is period 1, period 2, period now not all period doubling bifurcations lead to chaos so the sequence reverses back period 2 forward and then back to 2 and 1 and so forth so only in one dimensional maps which have certain characteristics called Schwartz's derivatives of certain kinds that the sequence is complete otherwise nonlinear dynamical systems it can so there are two branches in this branch sequence is not complete it just goes from p1, p2, p4 back to p2 and so forth so this is basically if you keep exciting at different frequencies but there is also another branch the sequence does go into chaos so here are some periodic orbits in this branch and you can see period 1, period 2, period 4 and so forth so you have period doubling bifurcations limit cycles chaos to various types of orbits something called Rossler, Lorentz, homoclinic orbit, Selnikov and so forth everything can be seen sorry yes they were known this is around 1990 89, 80 so this is a PID students work starting in 87 till 90 yes but again they are only for one-dimensional maps logistic maps and so on four-dimensional conservation yes well that was the motivation can I see the route to chaos as a simple period doubling bifurcation of course no there is Selnikov method route to chaos as well taking place for Selnikov behavior you have to show the existence of homoclinic orbits and then certain eigenvalues of the fixed points ratios and so on criteria can be built so here is a for the single periodic period solution branch you can see how the period of the solution is going to infinity so somewhere at this value of the frequency there is a homoclinic orbit and if you study some more carefully the study properties of that homoclinic orbit you can show that there is a Selnikov behavior okay so again there is this Rossler type solutions they merge into Lorentz type solutions and so on there was you know in those days fancy words were being created chaotic attractors of course something called crisis crisis is a phenomena in which some attractor gets destroyed by its touching the basing boundaries of or the stable manifolds of some other unstable equilibria now detecting it is a trick or painstaking process so we were able to detect in the case of the string motions there is the chaotic behavior and then if I slightly change the parameter frequency here that this attractor is destroyed so destroyed means I start here for a long time it keeps going like this suddenly it comes very close to the stable manifold of this saddle point and that goes to here okay as I said lots of problems exhibit similar behavior okay so here is in spherical pendulum this is what the response curve looks like and these are the you know non-planar branches and we showed that the same kind of period doubling bifurcations and so on takes place here similarly motion of the beam so and this is just evidence of some of that so there is time up I think I have three four more minutes and now I want to give people the opportunity to ask questions okay so let me talk a little bit about order reduction in dynamical systems of course here I did already order reduction everybody saw I started with infinite number of modes of the beam I only kept two modes well at least asymptotic theory I could show that all the other modes decay if there is sufficient damping they all go to zero okay because they are not being directly excited either through external forcing or through coupling with the other modes due to non-linear coupling in the case of strings it is relatively easy because all the frequencies are in multiples but in the case of beams and so on whether higher order resonances arise which are called combination resonances and so on that's not so easy to prove analytically okay so in most non-linear system the idea is still looking for can I reduce the problem size of the problem so I can study a small order system predict something about it because I can study smaller order systems more easily and then conclude something about the larger system okay so again looking at using singular perturbation theory externally excited system with multiple well potentials in those days it was fashionable to study multiple well potentials show chaos and so forth and Malnikov analysis for one degree of freedom or two degree of freedom systems could be done and for analysis for higher order systems is quite difficult okay so how can we arrive at reduced order systems if the reduced order system exhibits chaotic behavior does the original system also done okay so I worked with Martin Korles in Aero and we use singular perturbation theory and Giorgio was the PhD student and we took this starting example so it's a two degree of freedom system this mass and this mass this base is excited with harmonic oscillation so we could reduce it to a two degree of freedom system introduce a to a small parameter which is the ratio of natural frequency of one or the other omega one over omega and this if mu is small one could show that there are two oscillators the two oscillators one has low frequency the other is high frequency and so one can call soft oscillator and stiff oscillator okay and so the motion equations become something like this x dot equals this is the soft oscillator this is the stiff oscillator and then there is a theory you can say existence of slow manifold and fast manifold and so forth in the end with the bottom line so we get motion on slow manifold and say if I can capture motion on slow manifold the fast manifold is a slave it follows the slow manifold whatever is happening on the slow manifold right it's not zero but okay so here is the response of the system predictions if the system has periodic motion so there are two curves here you can see one is the original system two degree of freedom system the other one is the single degree of freedom reduced order model and you can see how they faithfully produce depending on the value of the parameter mu well this is not chaos these are just periodic orbits so here is the chaotic orbit okay so this is for 0th order approximation and this is for the full system so the chaotic orbits look relatively quite good so we could prove then that one degree of freedom system does represent the full order systems behavior reasonably okay alright so I don't want to leave this this is an area in which seat occupant dynamics Professor Davies, Patricia and myself have spent now somewhere around 17-18 years okay work with support from Johnson controls and NSF and so on so at least I wanted to give you a flavor of what that is so foam is a complicated material so it represents creep and it has long-term and short-term constants significantly then if I just take a foam block and compress and uncompress it shows the significant amount of hysteretic behavior and I'm not talking about small amplitude compression so it's like I take a 3 inch block reduce it to 1 inch so it's around 70 percent 67 percent and then bring it back to 3 inch and so this is the compression versus force and you can see that there is I can't call something as a linear stiffness of the foam there is significant amount of non-linearity in it okay and this is now we were working with 3 different types of foam in the car seat this was the Dodge car seat and so the blue we were calling soft foam and then intermediate foam and red is stiffer foam okay so here is some simple modeling of a car seat now of course we can do finite element modeling it's not so straightforward there are lots of different issues right so first thing we wanted to predict can we predict if I put a dummy in the car seat where does this dummy settle down in the car seat and the car seat has foam seats and seat back okay and so can we do that and the foam is highly non-linear material depends on temperature and humidity so if your car is parked in Arizona versus in Alaska the occupant is going to settle in different positions in the car seat okay alright so this settling point is called H point location and it also determines what the pressure distribution between the occupant and the seat is so those are important things right and then dynamically if you excite the seat then it's called these seat to head transmissibility and apparent mass and all that stuff okay so just as a caricature here is an occupant sitting on the car seat what happens if I take the occupant and put him on this car seat so I just put the occupant by these three rods well these rods start deforming and they are only linked at the joints okay so rigid bodies so that's how something like this takes place and so the static equilibrium position this is how one can determine some sort of a of the and this point is called the H point of this car manufacturers give this H point because that's what determines the location of all the controls in the car seat okay so that's what they give to the car seat manufacturers and say now produce a seat which meets the specifications relative to the interior of the car okay as you can see immediately this will depend on what not only the firm who the occupant is a 25 year 25 pound kid versus a 500 pound person okay the equilibrium positions will be quite different okay so the modeling framework we developed a modeling framework which the main elements of the seat occupant system different physics based on the models seating firm characterized by constitutive non-linear firm material as visible last thing the other complicated thing is interfaces the occupant sits on the seat while occupant slides relatively not glued together so we can't say it's a multi-degree of freedom system like this anyway so we try to develop a model so you can see this is the initial position of the occupant and after 10 seconds it has gone to something right so this model has some prediction capabilities let's see and of course I should have thought about it right because I copied the slide onto the stick there went my yes okay anyway learn lessons so here is an experiment with a mat right so you can have pressure mat and so in the invasion center with the dummy this experiment was done and this is the pressure distribution underneath the occupant and so if you do some post processing this is how the pressure distribution looks like along the center line of the occupant our model is planar so it's only on the center line and so this is what our model prediction looks like not too bad with all the approximations and so on we have improved on this significantly now with additional modeling efforts and so on okay so let me spend 5 minutes on my role in the administration of the surveys so I became graduate chair in 1998 courtesy of and that was before he had announced that he was going to noted him as the dean okay at 2 months before that okay then when Dan Harleman came he decided to upgrade the position to associate head for graduate education and research was the first chair of the research committee and so that was in 2001 okay and then so these are some of the things I was privileged to be able to do with our faculty went from around 238 students to around 515 students in that period directed developed a direct PAD program PAD enrollment went from around 30% to 60% in the school developed a combined BSMS and helped do some of these things okay then I became the interim head of the school in 2010 September August September and in May 2011 I was appointed the head of the school the greatest achievement or contribution to the school hopefully positively okay only time will tell is the people that I recruited to the faculty right and so you can see there is I thought I will not put the names and make an effort to try to remember the names right so this is Tahira this is Jitesh Rebekah Dave this is Amy this is Pavlos that's Marcial that's Arzu this is Iliad this is Guillermo that's KG that's Karlo that's Pham that's Benchin that's Neera this is Fabio this is Andres that's James that's Ivan that's Terry that's Chris that's Adrian that's Liang and that's Karlo right one is that twice I kept going back and forth between the two slides to see am I repeating anybody so there are eight in each line so that's 23 tenure track full time in Amy and there are the four joint appointments that's Liang Guanglin and that's Ed and that's Chiwan and that's our friend he will he was really as visiting assistant professor and will become the regular okay so here are the collaborators they are quite of course so many years 35 years here lots of people have touched my life of course I don't remember everyone who has touched my life some people I remember very fondly I remember when I was going up for promotion for full professor Professor Henry Yang he was the dean he called me in his office and said don't hear to any rumors that you won't be promoted okay and I just was called by so I don't know how many of you know what the process is for promotion to full professor or even to for any promotion the university promotion committee has these 90, 100, 120 applications for promotion from across campus the members of the committee read and they vote and those who clear two thirds vote they are not discussed in the university panel only those and then those who receive less than one third are not considered in the university panel the one between one third and two third they are considered by the university panel unless somebody member says I would like to discuss this so and so case other than those that's the so I guess my case was in that one third to two third case and Henry Yang was asked to say something about I or not and so on anyway I don't know the rest of the story okay quite a few colleagues who have helped me of course Osita is no more he passed away in 1997 or 1998 or so 99 lots of graduate students who have helped and there are many sponsors who have supported some of this research okay I missed out on many people and many areas of research okay so I studied Eigen value veering and mode localization with Osita break squeal modeling and prediction with Chuck flow induced vibrations with Irwin nonlinear MEMS modeling uncertainty analysis with Marisol and Irwin optimal design of nonlinear MEMS and uncertainty quantification and so on thank you I think our host is Malva so the car seat phenomena you demonstrated is very fascinating now what happens is we have ability to change car seat height and backside so the edge factor is changing accordingly yes so what happens is we did some research on that so basically where even where you place the occupant I enter the car I don't exactly sit at the same place so my location in the car seat 2 inches front 2 inches back or I go and hit the seat back and so on all that changes the edge point now it's not that critical how much edge point gets located it's really the comfort level of the occupant the more important other thing is how does it affect then the dynamics of the ride and as you travel on the car in the car and whether you are doing short trips or long trips and so on because that affects your fatigue and so forth so we have the capability now the models to really do all the dynamics and predict all those quantities yes sir so are you a contributor or are you guilty of making any of those names that got real popular in the early 90s like the chaotic whatever and I mean interesting names a crisis pointer I can't remember them but I did not create those names I did not create those names no I'm not surprised why not capable of creating those names no no I didn't say that I'm not sure you could have I usually I hate to reproduce I hate to give my own names to things that people have already and I don't like people who don't look at the literature explore the literature before coming up with their own names and I know of many such things that bother me yes and then the technical meaning should be the same for all things the same like you know chaos but the hell is chaos right so I feel quite agitated when lots of people use the word chaos without explicitly clearly stating what do they mean by do they only see a wiggly curve and they call that a chaotic looking trajectory okay once they add the objective looking okay I can live with it but if they say chaos and you can find lots of such things yeah okay alright thank you very much colleagues my pleasure