1 00:00:16,120 --> 00:00:21,370 Welcome to all of you watching streaming or catch up. It's a great pleasure. I would say to be back 2 00:00:21,370 --> 00:00:27,070 in Oxford where I started, of course, it was in a different building in those days. But I have been in this building 3 00:00:27,070 --> 00:00:32,140 a number of times before, and it's a great honour and privilege to give this lecture today. So I 4 00:00:32,140 --> 00:00:37,840 slightly change the title. I wanted to call it on the Millennium Bridge Synchronisation Myth. 5 00:00:37,840 --> 00:00:42,850 And the subtitle is Why Pedestrian Bridges Wobble Synchronisation and the 6 00:00:42,850 --> 00:00:48,340 Wisdom of the Crowd and Hear. The Wisdom of the Crowd refers to how crowds behave on 7 00:00:48,340 --> 00:00:53,770 moving structures and how crowds behave generally, but also how the wisdom of the crowd 8 00:00:53,770 --> 00:00:58,990 can sometimes, in my view, lead to scientific theories and something that inverted 9 00:00:58,990 --> 00:01:04,000 commas. Everybody knows, don't they? That might not be true or might not be as simple as 10 00:01:04,000 --> 00:01:09,160 that. And I should say at the outset, this is work with my colleague John McDonald in civil 11 00:01:09,160 --> 00:01:14,170 engineering and his former APHC student, Matt Bostian, who's now at the University of 12 00:01:14,170 --> 00:01:19,300 Leicester, and also Igoe Belic, Russell Jita and Kevin Dailey, who were in Georgia 13 00:01:19,300 --> 00:01:24,730 State University. So you may have seen, I don't know, 20 14 00:01:24,730 --> 00:01:29,830 years ago. It is now almost 20 years ago, you may have seen videos 15 00:01:29,830 --> 00:01:34,960 or you may have been present at what happened on so-called London Millennium Bridge. Now, 16 00:01:34,960 --> 00:01:40,210 the London Millennium Bridge was billed as the world's first horizontally sprint 17 00:01:40,210 --> 00:01:45,460 suspended suspension bridge. And it was. It links the Tate 18 00:01:45,460 --> 00:01:50,800 Modern Gallery was in Paul's Cathedral. It's an iconic structure. And it was billed 19 00:01:50,800 --> 00:01:56,170 as it won awards because of its unique heritage or its unique design 20 00:01:56,170 --> 00:02:01,360 as a collaboration between an architect and an engineer. But on the day 21 00:02:01,360 --> 00:02:07,200 it opened, this happened. The Millennium Bridge is a stunning aluminium and steel structure. 22 00:02:07,200 --> 00:02:12,310 A bit of this is a BBC programme. All of my videos in Italian, it's night and 23 00:02:12,310 --> 00:02:17,360 day, 10th of June 2000. But as soon as 24 00:02:17,360 --> 00:02:22,460 the crowds hit the bridge, there was a problem. It wobbled and not just a bit, but enough to 25 00:02:22,460 --> 00:02:27,740 make people clinging to the handrails for security. Clearly, the crowds themselves 26 00:02:27,740 --> 00:02:33,770 were driving the bridge in a completely unexpected and very worrying way. 27 00:02:33,770 --> 00:02:39,350 The bridge was closed three days later and it's remained closed ever since. But how could this have happened? 28 00:02:39,350 --> 00:02:44,990 Why wasn't it? How did it happened? Well, I want to go back into the said the popular theory. 29 00:02:44,990 --> 00:02:50,030 Roughly speaking, is when you look at that video, I don't know what you saw, but you might 30 00:02:50,030 --> 00:02:55,130 have seen a whole lot of heads that appeared to be moving in synchrony. And 31 00:02:55,130 --> 00:03:01,280 there is a strong belief that what happened was that as people started to synchronised their footsteps. 32 00:03:01,280 --> 00:03:06,500 So the bridge started to have its natural frequency amplified and as its 33 00:03:06,500 --> 00:03:11,630 frequency was amplified, so that synchrony increased. And this 34 00:03:11,630 --> 00:03:17,240 is used as almost a I would say, a sort of a billboard poster or a canonical example 35 00:03:17,240 --> 00:03:22,340 of the mathematical theory of synchronisation. It was very convenient because the year 2000 was when that 36 00:03:22,340 --> 00:03:27,860 theory was really taking off. So I want to tell you about the mathematical theory of synchronisation. 37 00:03:27,860 --> 00:03:32,960 And it goes back to Christian Horgan's who until did Cobb Parkway Station. 38 00:03:32,960 --> 00:03:38,060 I didn't realise had two A's in his Christian name. But as always, I'd been preparing these 39 00:03:38,060 --> 00:03:44,090 slides right at the last minute and so called Hagans clocks. 40 00:03:44,090 --> 00:03:49,250 Then there are lots of other examples of synchrony. And you if what I'm saying 41 00:03:49,250 --> 00:03:54,440 is boring or not interesting, you can look at these cats and decide whether they are synchronising 42 00:03:54,440 --> 00:03:59,600 or not. Also, brain waves, power grids are power grids rely on 43 00:03:59,600 --> 00:04:04,790 the idea that the whole of the AC circuit that supplies power to the whole 44 00:04:04,790 --> 00:04:10,460 of a single grid like the nation is completely in synchrony. So, you know, that AC circuit 45 00:04:10,460 --> 00:04:15,650 is essentially well, you can think of it as a wave vector of polarised 46 00:04:15,650 --> 00:04:20,900 electromagnetic radiation, but that has to be completely in sync across the whole network. Otherwise 47 00:04:20,900 --> 00:04:26,370 it goes down and rhythmic clapping. Well, how does that come about? We'll talk about that 48 00:04:26,370 --> 00:04:31,820 then. I want to talk about Yoshiki Kuramoto, who came up with a really simple 49 00:04:31,820 --> 00:04:38,990 mathematical theory of synchronisation, which I will show you. At least I will do a bit of maths on the board 50 00:04:38,990 --> 00:04:44,000 and then give you a hint. How you can go from two to many oscillations 51 00:04:44,000 --> 00:04:49,040 and then give you the explanation for them under a millennium bridge. But really, this is going to be a tale 52 00:04:49,040 --> 00:04:54,230 of two bridges, because there's another theory of how 53 00:04:54,230 --> 00:04:59,330 the London Millennium Bridge and other footbridges might go unstable. And I might call that theory. 54 00:04:59,330 --> 00:05:05,030 How not to fall over. Quite a simple theory. It's a bit like Douglas Adams, How to Fly, 55 00:05:05,030 --> 00:05:10,040 which is aim for the ground and then miss. 56 00:05:10,040 --> 00:05:15,320 And this was actually triggered, amongst other things, by my colleague John MacDonald's observation on another iconic 57 00:05:15,320 --> 00:05:20,540 bridge, the Bristol Clifton Suspension Bridge. And I'll explain to you his model 58 00:05:20,540 --> 00:05:26,630 of walking and the idea of so-called negative damping, averagely speaking 59 00:05:26,630 --> 00:05:31,670 and then simulating and some results from some three simple models 60 00:05:31,670 --> 00:05:36,770 and then ask what is the right answer? And perhaps that's always a difficult question. When you're doing maths of the real world, there 61 00:05:36,770 --> 00:05:42,080 is no such thing as a perfect answer. When you doing mass of the real world. And I might 62 00:05:42,080 --> 00:05:47,120 give some philosophical remarks or I might run out of time. I certainly 63 00:05:47,120 --> 00:05:52,310 will run out of time because as I was telling Diarrhoeal earlier, who runs these? My general 64 00:05:52,310 --> 00:05:57,620 way that I give lectures and, you know, I get student feedback that says enthusiastic. 65 00:05:57,620 --> 00:06:03,090 Yes. Five out of five. Did we understand it all? Can we follow it for the exam? Yeah, 66 00:06:03,090 --> 00:06:08,330 you did. What was the best thing about Professor Champion is lecture the first time he told the joke. 67 00:06:08,330 --> 00:06:13,340 What was the worst thing about his lecture? The sixth time he told the joke. So I tend to 68 00:06:13,340 --> 00:06:19,130 prepare meticulously and then say the first thing that comes into my head. So I'll almost certainly 69 00:06:19,130 --> 00:06:24,280 go over time, for which apologies, but I will speed up. 70 00:06:24,280 --> 00:06:29,600 So Christian Horgan's was a sort of contemporary of Newton, and he's a physicist, 71 00:06:29,600 --> 00:06:35,540 a mathematician, but also an inventor. And he described it discovered by chance 72 00:06:35,540 --> 00:06:40,580 an unexpected or odd sympathy that he observed at home. And 73 00:06:40,580 --> 00:06:45,740 here's a diagram of what he had. He had two pendulum clocks suspended, but on a 74 00:06:45,740 --> 00:06:54,980 rack between two shelves. And people have done modern versions of this. 75 00:06:54,980 --> 00:07:00,020 So these are not pendulum clocks. These are metronomes. They're not quite tuned to the same 76 00:07:00,020 --> 00:07:05,060 frequency and they're not in sync. And what we're now going to do is we're gonna couple them 77 00:07:05,060 --> 00:07:10,160 or this physicist is going to couple them. This YouTube is going to couple them by 78 00:07:10,160 --> 00:07:15,170 putting them on a plank, rather like I have there, and watch what happens or listen 79 00:07:15,170 --> 00:07:24,510 to what happens. 80 00:07:24,510 --> 00:07:29,520 Just a tiny coupling between the pendulum's by the fact that 81 00:07:29,520 --> 00:07:35,010 they are all uniformly contributing to the same sway of this plank, 82 00:07:35,010 --> 00:07:40,200 that they're all coupled together through the vibration of the plank causes them 83 00:07:40,200 --> 00:07:45,420 all to fall into sync. And the plank to start to move. 84 00:07:45,420 --> 00:07:51,960 Oh, no, I don't want to do that. Thank you. And in fact, this idea of synchrony 85 00:07:51,960 --> 00:07:57,030 has been observed in many other situations in the 86 00:07:57,030 --> 00:08:02,730 physical world. So this is a picture this isn't a video because I couldn't find it of 87 00:08:02,730 --> 00:08:08,370 that. I took in the gift shop of the Great Wall of China. I've only been to China twice. This was the first time. 88 00:08:08,370 --> 00:08:13,440 And there was the conferences in Beijing and there's the obligatory afternoon expedition 89 00:08:13,440 --> 00:08:18,510 to the Great War. But it tipped them frankly, it was absolutely pouring. I'm doing my 90 00:08:18,510 --> 00:08:23,550 best not to swear. I very nearly said a rude word, but it was it was raining cats and dogs. Fortunately, 91 00:08:23,550 --> 00:08:29,310 it cleared up and we did get to go on the Great Wall, but there was about 10 or 15 of us mathematicians and physicists 92 00:08:29,310 --> 00:08:34,320 who became fascinated by this display of waving cats. And you might be able 93 00:08:34,320 --> 00:08:39,830 to see it in this still here. You might just be able to see it or there's a point you might just be able to see 94 00:08:39,830 --> 00:08:45,000 that if you look at each row is in sync with each other. This row here is 95 00:08:45,000 --> 00:08:50,040 in there all in sync with each other. This row here is approximately in sync. And this row here 96 00:08:50,040 --> 00:08:55,110 is approximately in-sync. And we started playing games. If what if we take one of the cats out? What if 97 00:08:55,110 --> 00:09:00,240 we move one of the cats so that it moves its arm? What if we swap and turn them backwards, etc.? 98 00:09:00,240 --> 00:09:05,250 Great fun for mathematicians and physicists, probably of no relevance whatsoever. These are 99 00:09:05,250 --> 00:09:14,870 fireflies. 100 00:09:14,870 --> 00:09:21,470 And after a while, looks towards the end of that clip. I hope you can see 101 00:09:21,470 --> 00:09:26,750 that they're beginning to synchronise. Todd, find yeah. Here. 102 00:09:26,750 --> 00:09:32,000 After a while, when there's a sufficient density of them, they start to synchronise. This 103 00:09:32,000 --> 00:09:37,580 is what happens in Powergrid. So this is an example of two electrical generators 104 00:09:37,580 --> 00:09:42,650 that are being made, being driven at slightly different speeds, slightly different voltages, 105 00:09:42,650 --> 00:09:47,660 and then an electrical coupling between them is switched on. And watch what happens. So these are 106 00:09:47,660 --> 00:09:54,500 two different electrical generators. 107 00:09:54,500 --> 00:09:59,570 So there's a strobe on this. I should have given a trigger warning. I'm very sorry. Look away if strobes are a problem. But it's 108 00:09:59,570 --> 00:10:05,230 not the whole room. It's just the video. 109 00:10:05,230 --> 00:10:10,270 So he's now going to switch on an electrical coupling, they're just sharing. They're just going to be part of the same 110 00:10:10,270 --> 00:10:15,370 circuit and being part of the same circuit is going to be enough 111 00:10:15,370 --> 00:10:21,400 when he switches on that coupling. Now, 112 00:10:21,400 --> 00:10:26,880 he hasn't changed the drive to either machine, but they synchronise to the common rotation 113 00:10:26,880 --> 00:10:32,040 frequency and the common face. It's just a weak 114 00:10:32,040 --> 00:10:37,170 electrical coupling between them. Whoops. 115 00:10:37,170 --> 00:10:58,930 We go back and this is a video of an audience clapping. 116 00:10:58,930 --> 00:11:04,330 How does that happen? Was there somebody and the conductor there wasn't on the stage going? 117 00:11:04,330 --> 00:11:09,370 Now, now, now, now. So not only did they all hit the same 118 00:11:09,370 --> 00:11:14,450 frequency, but they all hit the same frequency and the same face. In fact, there has been 119 00:11:14,450 --> 00:11:19,450 a very interesting study recently of somebody looking at this. And they discovered one of the things that 120 00:11:19,450 --> 00:11:24,490 you might notice about the two halves of that clip. I'll show it again. It's hard to see 121 00:11:24,490 --> 00:11:29,500 this because you're seeing the collective clapping in the beginning, but actually it's hard 122 00:11:29,500 --> 00:11:34,780 to synchronise your clapping if you clap too quickly. Generally, clapping is liable 123 00:11:34,780 --> 00:11:40,000 to be synchronised if you stop to clap more slowly. And they discovered 124 00:11:40,000 --> 00:11:45,130 in communist countries that when the there was a fantastic 125 00:11:45,130 --> 00:11:50,410 speech that everyone enjoyed. There was a lot of clapping when the Dear Leader gave a speech 126 00:11:50,410 --> 00:11:55,600 that they were obliged to clap. Suddenly there were synchrony. Now you draw your own conclusions 127 00:11:55,600 --> 00:12:08,870 about why that might be. But listen to this again. Just watch an individual. They're clapping quite quickly. 128 00:12:08,870 --> 00:12:14,230 Quite enthusiastically. Slightly less enthusiastic. And it slows 129 00:12:14,230 --> 00:12:19,320 down into synchrony. That's interesting. I don't quite know why. So what 130 00:12:19,320 --> 00:12:24,360 am I going to talk about? Well, as I was sort of intimating there, there's two ways you can make 131 00:12:24,360 --> 00:12:29,430 independent oscillatory things oscillate. There's the orchestra kind of way. Orchestras 132 00:12:29,430 --> 00:12:34,440 are in sync because there's a conductor, there's a conductor who says now everybody make the 133 00:12:34,440 --> 00:12:39,600 same down string on your bow or for the violinist. There's a leader of the orchestra. There's a follow my 134 00:12:39,600 --> 00:12:44,790 leader. But there's also a spontaneous synchronisation, like much like 135 00:12:44,790 --> 00:12:49,830 you might get an ensemble piece. And I'm gonna talk about this spontaneous kind 136 00:12:49,830 --> 00:12:54,930 of synchronisation because there was certainly nobody on the Millennium Bridge. You do get synchronisation, by 137 00:12:54,930 --> 00:13:00,030 the way, when you play music. Another way of getting synchronised clapping is during 138 00:13:00,030 --> 00:13:05,100 rock concerts. And in fact, there can be problems in when rock concerts are played 139 00:13:05,100 --> 00:13:10,620 in sports stadia because sports stadia are designed to take a certain weight of 140 00:13:10,620 --> 00:13:15,690 crowd. What they're not designed to take is a certain weight of crowd who are all doing this 141 00:13:15,690 --> 00:13:20,820 at the same time in the stands. And there've been examples where huge instabilities have been 142 00:13:20,820 --> 00:13:26,880 set up due to orchestrated synchronous jumping. 143 00:13:26,880 --> 00:13:31,950 So the spontaneous case of the fireflies. There's nobody telling the fireflies when to turn off. There's 144 00:13:31,950 --> 00:13:37,050 nobody telling you when to clap. The metronomes. There was nothing to tell the metronome which 145 00:13:37,050 --> 00:13:43,810 face to choose. And these even cats that may or may not be synchronising. 146 00:13:43,810 --> 00:13:49,680 There's nothing to tell them what face to take. And there's a fundamental theory 147 00:13:49,680 --> 00:13:54,900 due to Kuramoto of phase coupled oscillators. And his very clever 148 00:13:54,900 --> 00:14:00,120 idea is consider n. Mathematicians like to call things 149 00:14:00,120 --> 00:14:07,150 and or Upsilon or whatever. So N is some large number, large being two or more. 150 00:14:07,150 --> 00:14:12,210 That's a private joke. But then imagine that they're coupled in 151 00:14:12,210 --> 00:14:17,400 some way. Perhaps they're too all coupled but weakly. And I'm not going to say what week means. I'm gonna deliberately 152 00:14:17,400 --> 00:14:22,830 say I'm not going to say what coupling is. But they're all things that want to oscillate and they want to oscillate. 153 00:14:22,830 --> 00:14:27,870 And I don't care how they oscillate, whether it's backwards and forwards or whether it's round and 154 00:14:27,870 --> 00:14:32,880 round. But I'm going to describe their oscillation by its phase. In other 155 00:14:32,880 --> 00:14:37,960 words, it's fraction of a period. So I'm not I'm not I'm going to throw away all the 156 00:14:37,960 --> 00:14:43,570 physics of simple harmonic motion or whatever, and I'm going to suppose that these are things that are naturally 157 00:14:43,570 --> 00:14:49,300 oscillating. So I can represent that motion in some sense as a circle of phase. 158 00:14:49,300 --> 00:14:54,730 I go from zero round to 360 degrees, et cetera. 159 00:14:54,730 --> 00:15:00,310 And so how we might write that mathematically is I describe this phase variable Phi 160 00:15:00,310 --> 00:15:05,440 or Phi I and defy D.T. Newton invented 161 00:15:05,440 --> 00:15:10,810 this dot notation for rate of change. That means the rate of change is just 162 00:15:10,810 --> 00:15:16,210 Omega. What's Omega? Omega is the frequency. So in other words, I'm just something that is constantly changing 163 00:15:16,210 --> 00:15:21,220 my face at a constant feed speed omega and then going back 164 00:15:21,220 --> 00:15:27,060 again and starting again. So I'm going round and round, if you like, at a constant speed. 165 00:15:27,060 --> 00:15:32,230 And then I'm going to suppose that I'm listening. In some sense, and I don't know how that listening is working, 166 00:15:32,230 --> 00:15:37,360 it may be listening through my ears. It may be listening through the vibrations in the plank. It may be magnetic with those 167 00:15:37,360 --> 00:15:42,880 cats, I'm not sure. It may be through current and voltage, but it's kind of weak. 168 00:15:42,880 --> 00:15:48,160 There's some weakness that says if my neighbour or something that I'm coupled 169 00:15:48,160 --> 00:15:53,380 to has a different phase to me, I'm going to adjust. There's some feedback 170 00:15:53,380 --> 00:15:58,420 that says if we're not in the same phase, I'm going to sense that I don't know what that feedback 171 00:15:58,420 --> 00:16:04,390 is. It's some function. I'm just going to call F and there's some coupling strength kapper, 172 00:16:04,390 --> 00:16:09,600 which is this weakness. So I'm I'm. This if you look at these cats, the phase 173 00:16:09,600 --> 00:16:14,740 is how far through the cycle of backwards and forwards I am. And Cappa is the extent to which 174 00:16:14,740 --> 00:16:20,350 I am being influenced by one of my neighbouring cats or one of the further away cats. 175 00:16:20,350 --> 00:16:25,650 So these are the small couplings and they're typically very much less than one. 176 00:16:25,650 --> 00:16:31,030 And if you think about it hard enough, you realise this only makes sense in this phase coupled sense, 177 00:16:31,030 --> 00:16:36,340 if the function F is symmetric, if it's to pay periodic, and 178 00:16:36,340 --> 00:16:41,710 I assume that if there's no phase difference, then these things are just happy to oscillate. 179 00:16:41,710 --> 00:16:46,960 So F of zero is zero and they realise the simplest functions that do that 180 00:16:46,960 --> 00:16:52,030 are just simple sine waves. So the simplest thing you can take if 181 00:16:52,030 --> 00:16:57,940 you know about how to decompose functions that or signals that aperiodic 182 00:16:57,940 --> 00:17:02,980 into sine waves, you would say you take a one mode Fourier truncation. You just you 183 00:17:02,980 --> 00:17:08,050 just take the simplest possible function. You just take the simplest possible law. I don't care what it is. I'm just 184 00:17:08,050 --> 00:17:13,060 going to take the first term in some expansion. And that turns out to be a 185 00:17:13,060 --> 00:17:18,310 sine wave. What is the sine wave sign in here is just a 186 00:17:18,310 --> 00:17:24,010 sine wave is just something that does this. This is a sign of PHI 187 00:17:24,010 --> 00:17:31,890 from zero up to 360 degrees. 188 00:17:31,890 --> 00:17:37,140 As a function of PHY, so I'm just going to assume that I have these weekly coupled oscillators 189 00:17:37,140 --> 00:17:42,450 and in the case of all tool coupling where the coupling strength is the same, then Kuramoto 190 00:17:42,450 --> 00:17:47,520 first wrote down these equations and you can do really interesting things with these equations. What you find, I'm 191 00:17:47,520 --> 00:17:52,590 going to do a bit of maths. You're going to do a bit of maths with me, I hope. Let's take the case of 192 00:17:52,590 --> 00:17:57,940 N is two. OK. So I'm gonna take two oscillators. 193 00:17:57,940 --> 00:18:03,130 And I'm going to assume that this kapper is just some constant and on the previous slide. 194 00:18:03,130 --> 00:18:08,350 I had a cappa over end just to give you some scaling, but that's not strictly necessary. So, in fact, 195 00:18:08,350 --> 00:18:16,010 this kapper is kapper over two from the previous slide. Woops, sorry. In fact, it's two cappa. 196 00:18:16,010 --> 00:18:21,290 No, it isn't, it's kapper of it. Oh, sorry. Alan, shut up. Sorry, I didn't say that aloud, did I? No, 197 00:18:21,290 --> 00:18:26,370 glad. And let Delta Omega be the difference between 198 00:18:26,370 --> 00:18:31,410 these two frequencies. So these are these two generators that are going round fires is the angle of 199 00:18:31,410 --> 00:18:36,600 that generator and they're going round it very slightly different speeds. 200 00:18:36,600 --> 00:18:41,700 And now let me take the difference. Let me take the difference. And I'm just going to subtract these two 201 00:18:41,700 --> 00:18:47,640 equations if I just subtract these two equations. I get the following. 202 00:18:47,640 --> 00:18:52,710 After a tiny bit of algebra, I find that the difference between them, 203 00:18:52,710 --> 00:18:58,350 the psi dot. So that's five one minus five to. I subtract 204 00:18:58,350 --> 00:19:04,080 these two equations. I find this side dot is just the difference in the Omega's 205 00:19:04,080 --> 00:19:10,080 minus two times Kappa PSI. Then for this equation, I can prove some things. 206 00:19:10,080 --> 00:19:16,280 I can prove to you that if the coupling strength is large enough. 207 00:19:16,280 --> 00:19:21,440 And the frequency difference is small enough. Then this equation 208 00:19:21,440 --> 00:19:26,750 has two different equilibria. That is two different. Constant 209 00:19:26,750 --> 00:19:32,090 face solutions where both the face different sorry, the constant face, different 210 00:19:32,090 --> 00:19:38,090 solution, where the two oscillators will be in phase. And this is really easy to show because let me look at that equation. 211 00:19:38,090 --> 00:19:43,380 What is it? What does it say? It says fi dot, which is a right. 212 00:19:43,380 --> 00:19:48,440 Five dot is the rate of change of fi. 213 00:19:48,440 --> 00:19:54,140 As a function of FYE. Is. 214 00:19:54,140 --> 00:19:59,180 Delta Omega, let me assume Delta Omega is positive, it doesn't have to be. So that's the difference in these 215 00:19:59,180 --> 00:20:04,900 two frequencies. That's fairly small. Minus 216 00:20:04,900 --> 00:20:10,000 something to Kapper Times Sign of fire will sign is a graph that I 217 00:20:10,000 --> 00:20:15,820 had previously. So that's gonna be a function that looks something like this. 218 00:20:15,820 --> 00:20:20,860 And there's my 360 degrees. And there's my zero degrees. And that's badly 219 00:20:20,860 --> 00:20:27,040 drawn because this here should be at 220 00:20:27,040 --> 00:20:32,650 two kapper. Careful. Poor 221 00:20:32,650 --> 00:20:37,690 minus. Delta Omega. 222 00:20:37,690 --> 00:20:42,700 What have I done wrong? This is totally rubbish. This should go up like this. That should be 223 00:20:42,700 --> 00:20:47,710 a zero point. There should be a zero point and I should come back to there, and that should be my 360 224 00:20:47,710 --> 00:20:54,700 degrees. But notice what this does. If this. 225 00:20:54,700 --> 00:20:59,740 To Kapper minus Delta Omega, as I've drawn it, is 226 00:20:59,740 --> 00:21:05,360 less than one. If that down there is less than one, then I have to zero crossings. 227 00:21:05,360 --> 00:21:10,700 What happens at these zero crossings? Well, this graph tells me everything, because this 228 00:21:10,700 --> 00:21:16,860 if this is my angle, phy, this graph tells me whether five varies positively. 229 00:21:16,860 --> 00:21:22,800 Or five areas negatively, so here, defy duty is positive, which means if I start here, I go this way, 230 00:21:22,800 --> 00:21:28,510 I increase my face difference. But if I start if in this region. 231 00:21:28,510 --> 00:21:33,670 Defied E.T. The rate of change of FI or 5.0 is negative. So I go this way 232 00:21:33,670 --> 00:21:38,760 and I end up here. And similarly, if I start in this region 233 00:21:38,760 --> 00:21:43,890 because defied fighting is positive, I go this way and I'm going to 234 00:21:43,890 --> 00:21:49,090 end up here. And I go right round 360 degrees become zero 235 00:21:49,090 --> 00:21:54,130 degrees again and I'll end up here. But see what that does that shows me not 236 00:21:54,130 --> 00:21:59,140 only are there two zero crossings, fixed points, points of 237 00:21:59,140 --> 00:22:04,270 which fire doesn't change the rate of change of fire, zero. It shows me that this one 238 00:22:04,270 --> 00:22:09,610 is stable. Because if I start slightly to the right of it, I pop back, 239 00:22:09,610 --> 00:22:15,190 if I start slightly to the left, I pop back. And this one is unstable. 240 00:22:15,190 --> 00:22:20,290 If I start this side of it, I go all the way round the world and come back to here. And if I start 241 00:22:20,290 --> 00:22:25,330 this side of it, I come back to here. That's dynamical systems one oh one. We have done 242 00:22:25,330 --> 00:22:31,300 a stability analysis of a dynamical system. Just by looking at whether something is positive 243 00:22:31,300 --> 00:22:36,790 or negative. And that shows me that there is a state and we call this the synchronous 244 00:22:36,790 --> 00:22:41,980 state where the difference in fire is small. It's close to zero 245 00:22:41,980 --> 00:22:47,320 and it's constant. And this provided this Delta Omega divided 246 00:22:47,320 --> 00:22:52,390 by two K is less than one. If everything is positive, so provided 247 00:22:52,390 --> 00:22:57,670 if this condition here is true. Then 248 00:22:57,670 --> 00:23:02,800 I'm going to send tend to this in phase equilibrium rather than the outer phase one, 249 00:23:02,800 --> 00:23:07,810 and that's going to be stable. Now, you can generalise that analysis. That was an analysis with 250 00:23:07,810 --> 00:23:13,120 just two oscillators. If I have many for simplicity, let's suppose everybody 251 00:23:13,120 --> 00:23:18,280 interacts with everybody else. Then you can do some things statistically. You can write down what's called 252 00:23:18,280 --> 00:23:23,740 a master equation. You can assume that the whole ensemble 253 00:23:23,740 --> 00:23:28,780 of all of these things, on average, you can add together how they oscillate. 254 00:23:28,780 --> 00:23:33,970 And you can say that. You can say that if you look at how something oscillates and you look at its X 255 00:23:33,970 --> 00:23:39,640 component. So if I think of something that's going round in a circle and I look at it from the side, 256 00:23:39,640 --> 00:23:44,950 then it goes up and down like a sine wave. And if I look at it from the top, then it goes backwards and forwards 257 00:23:44,950 --> 00:23:50,260 like a cosine wave, which is the same as sine, but just shifted by 90 degrees. 258 00:23:50,260 --> 00:23:55,420 And that tells me that I can define if I average everything. Sorry, this 259 00:23:55,420 --> 00:24:00,880 don't worry about this, I. That just means I'm looking from the top or from the side effectively. 260 00:24:00,880 --> 00:24:06,280 But there's a neat way of writing that which is called Euler's formula that we don't need to worry about. But 261 00:24:06,280 --> 00:24:11,410 essentially I can do some averaging and say on average, I can think of this 262 00:24:11,410 --> 00:24:16,490 as a big oscillator. Multiplied by 263 00:24:16,490 --> 00:24:21,560 something that's going round a circle, and depending on whether that average 264 00:24:21,560 --> 00:24:26,600 is depending on whether that if the average is one, that 265 00:24:26,600 --> 00:24:31,740 tells me that everything. Is in synchrony if that average 266 00:24:31,740 --> 00:24:36,770 is zero. It tells me that they're all over the place. There is 267 00:24:36,770 --> 00:24:42,080 no there is no synchrony at all. And this is called an order parameter, and Kuramoto invented 268 00:24:42,080 --> 00:24:47,090 this order parameter that says how much of this is in synchrony 269 00:24:47,090 --> 00:24:52,370 and how much is it not in synchrony? And more recently, people have been interested in what happens 270 00:24:52,370 --> 00:24:58,160 in the boundary between synchrony and non synchrony. In the case where the coupling strength is 271 00:24:58,160 --> 00:25:03,500 not really large enough to cause complete synchrony. But it's not zero. There is some coupling. 272 00:25:03,500 --> 00:25:08,750 And these are being called Chimaera States. chimaeras are supposed to be creatures that are composed of two different 273 00:25:08,750 --> 00:25:14,150 organisms, like a lion with a with a donkey's 274 00:25:14,150 --> 00:25:19,460 head or something. And the idea is you can have these oscillator, some of which want to synchronise 275 00:25:19,460 --> 00:25:24,850 and some don't. So here are some simulations, again, just 276 00:25:24,850 --> 00:25:29,890 pilfered off the web of the case when the coupling strength is 277 00:25:29,890 --> 00:25:35,620 large enough and the case where it isn't. So in this simulation on the left, there's just 278 00:25:35,620 --> 00:25:40,840 a whole load of coupled oscillators with representing their phase, the degree of half hour round they've 279 00:25:40,840 --> 00:25:45,970 gone. And in this case, it's for purely illustrative reasons. They're 280 00:25:45,970 --> 00:25:51,850 not all represented on a circle, but so that you can see them all there draw. They're pulled out 281 00:25:51,850 --> 00:25:57,730 transversely to the circle. So some of them are further away and some of them are nera in. That's purely for illustration. 282 00:25:57,730 --> 00:26:02,860 So if you take some that a starting and random position 283 00:26:02,860 --> 00:26:08,960 as you allow them to oscillate. So they start to swarm together 284 00:26:08,960 --> 00:26:14,000 and very quickly all start to oscillate with the same face. So this would 285 00:26:14,000 --> 00:26:19,040 be a case where the coupling strength divided by the phase 286 00:26:19,040 --> 00:26:24,260 difference is large enough to cause all these oscillators to fall into sync. 287 00:26:24,260 --> 00:26:33,670 And here is a case where that's not the oh, sorry about the music. 288 00:26:33,670 --> 00:26:38,740 There's a case and you can watch this for hours where the coupling strength isn't quite large enough, but 289 00:26:38,740 --> 00:26:43,990 you see this patch of fairly synchronous behaviour. These these oscillators are trying 290 00:26:43,990 --> 00:26:49,060 approximately to follow each other. Now, I told you that when I was at the Great Wall 291 00:26:49,060 --> 00:26:55,420 of China, myself and some other delegates at this conference found these 292 00:26:55,420 --> 00:27:00,460 waving cats that were perfectly in synchrony. Unfortunately, this weekend 293 00:27:00,460 --> 00:27:06,200 when I went and bought these cats and thought, yeah, I know what I'll do, I'll put them on the. Oh. 294 00:27:06,200 --> 00:27:12,470 But then I realised what these guys are doing is really interesting. They're not synchronising, 295 00:27:12,470 --> 00:27:17,690 but they are interacting with each other. What seems to happen if we look at this guy in the middle? Guy 296 00:27:17,690 --> 00:27:22,940 being a gender neutral term. He speeds up and slows down as large 297 00:27:22,940 --> 00:27:27,950 amplitude. And then as he slows down, the next one gains a large amplitude. As he slows down, the next 298 00:27:27,950 --> 00:27:33,410 one does bounces back. Then this one's got large amplitude. Then this one's got large amplitude. 299 00:27:33,410 --> 00:27:38,630 Then this one's got large amplitude. There's a travelling wave going on. And this is one 300 00:27:38,630 --> 00:27:43,730 of the things oscillators can do. If they don't have enough coupling strengths to cause them 301 00:27:43,730 --> 00:27:49,700 to fall into synchrony, you'll get this travelling wave of behaviour of large and small amplitude. 302 00:27:49,700 --> 00:27:55,140 And that seems to be what's happening in the experiment that I tried to do. 303 00:27:55,140 --> 00:28:00,230 So. What does this got to do with bridges? 304 00:28:00,230 --> 00:28:05,480 Well, the accepted, if you like, the textbook example of 305 00:28:05,480 --> 00:28:10,830 synchronisation and the Millennium Bridge, the explanation goes as follows. 306 00:28:10,830 --> 00:28:15,830 You think of the bridge as having a lot of pedestrians in each of the pedestrians are approximately trying to 307 00:28:15,830 --> 00:28:21,440 walk at the same frequency, approximately. But as the bridge just 308 00:28:21,440 --> 00:28:26,570 slightly moves, just those slight movements of the bridge is enough to push them more towards 309 00:28:26,570 --> 00:28:31,640 synchrony. And that's synchrony is enough to then start to make the bridge move. That then 310 00:28:31,640 --> 00:28:36,650 is enough to start to couple everything together. And that's essentially what this video shows with 311 00:28:36,650 --> 00:28:42,020 metronomes. If I take a enough metronomes and put them on this again, nicked off the Internet 312 00:28:42,020 --> 00:28:54,060 and take a large bridge like structure eventually. 313 00:28:54,060 --> 00:28:59,370 Almost like soldiers moving with military precision. The fact 314 00:28:59,370 --> 00:29:04,980 that they're all couples through the collective motion of the bridge is enough to make them slowly 315 00:29:04,980 --> 00:29:10,320 fall into approximate synchronisation. Left, right, left, 316 00:29:10,320 --> 00:29:15,910 right. Left, right, left. Right. Company. Halt, 317 00:29:15,910 --> 00:29:21,190 etc. So the full explanation 318 00:29:21,190 --> 00:29:26,590 goes something like this. This was the world's first horizontally suspended bridge. Bridges are designed 319 00:29:26,590 --> 00:29:31,660 generally to be vertically suspended. Suspension bridges have cables that come from above. And a lot 320 00:29:31,660 --> 00:29:37,360 of effort goes into making sure that there aren't resonant frequencies in the 321 00:29:37,360 --> 00:29:42,850 vertical vibrations. And, for example, it's been known for a long time. There's a command 322 00:29:42,850 --> 00:29:48,160 amongst troops to break step when walking across a bridge. Because if troops were to march 323 00:29:48,160 --> 00:29:53,230 across a bridge all in sync with each other, it was known that that caused large vertical 324 00:29:53,230 --> 00:29:58,530 vibrations of the bridge. So it was believed that as long as it wasn't soldiers marching, 325 00:29:58,530 --> 00:30:03,580 that pedestrians, all of whom have random phases and walk randomly across a bridge, 326 00:30:03,580 --> 00:30:08,710 were never likely to cause a bridge to go unstable. And so it seems they 327 00:30:08,710 --> 00:30:14,110 don't in the vertical domain. But they appear to in the horizontal domain. 328 00:30:14,110 --> 00:30:19,180 So if you take a bridge that has a horizontal elasticity and this one had a very large one 329 00:30:19,180 --> 00:30:24,850 because it was horizontally suspended approximately, then this 330 00:30:24,850 --> 00:30:30,280 synchronisation of footsteps, given sufficient numbers, was enough to cause 331 00:30:30,280 --> 00:30:35,530 synchrony and therefore an instability. And it doesn't really matter. So this theory goes how 332 00:30:35,530 --> 00:30:40,540 you model the pedestrians because they're all oscillators. And Kuramoto theory just relies 333 00:30:40,540 --> 00:30:45,640 on face coupling. It doesn't rely on the particular details of the oscillators. And it looks a bit 334 00:30:45,640 --> 00:30:51,070 like having a lot of Horgan's clocks or Hagans metronomes. And this is the parodic 335 00:30:51,070 --> 00:30:56,140 I can never say that word. It's a paradigm example of synchronisation. For example, 336 00:30:56,140 --> 00:31:01,240 have a look at the book Sync. It's an excellent book. In Steenstra, Getz's 337 00:31:01,240 --> 00:31:06,430 book in Penguin. And only about a month ago, I heard Hanna Fry, who is also superb 338 00:31:06,430 --> 00:31:12,100 on six music in the Morning, talking about synchronisation and using the London Millennium Bridge example 339 00:31:12,100 --> 00:31:17,160 as a classic example of synchronisation in the real world. Ashley, 340 00:31:17,160 --> 00:31:22,770 the engine is Arop, were called into reengineer the bridge. The bridge was closed for two years. 341 00:31:22,770 --> 00:31:27,810 And what they did is they made empirical observations. They didn't understand all this theory of synchronisation. 342 00:31:27,810 --> 00:31:33,440 They just made empirical observations. And they noticed that on average, 343 00:31:33,440 --> 00:31:38,700 a pedestrian provides some lateral force and that force appears 344 00:31:38,700 --> 00:31:43,710 as damping. But it appears as damping with a negative sign. So they they they 345 00:31:43,710 --> 00:31:48,750 said we can think of each pedestrian as like a negative damper, whatever that means. And they drew 346 00:31:48,750 --> 00:31:53,940 a straight line that actually got their employees and volunteers to walk across the bridge when it was closed 347 00:31:53,940 --> 00:31:59,460 and took all this data and drew a straight line through it and found that there is a a 348 00:31:59,460 --> 00:32:04,620 negative damping coefficient, whatever that means. And what they concluded was the way to solve this problem 349 00:32:04,620 --> 00:32:09,930 is stick ugly whacking great dampers on the bridge. Stupid engineers didn't understand 350 00:32:09,930 --> 00:32:15,360 the theory. That's kind of not what anyone's ever said. But what mathematicians and physicists sometimes 351 00:32:15,360 --> 00:32:20,430 think. I can remember being told as an undergrad and a maths degree. Come on. Think you're 352 00:32:20,430 --> 00:32:27,850 thinking like an engineer. As if that was kind of empirical and not deep enough. 353 00:32:27,850 --> 00:32:33,260 But. If you look at this was also from last year. Another fantastic 354 00:32:33,260 --> 00:32:38,480 lecturer, Matt Parker. He describes himself as a stand up mathematician. If you've ever seen him 355 00:32:38,480 --> 00:32:43,610 or you'll know what I mean. But if you haven't, I strongly recommend seeing one of Matt Parker's lectures. This 356 00:32:43,610 --> 00:32:48,800 is just last year in the Royal Institution and a lecture entitled When 357 00:32:48,800 --> 00:32:53,990 Mathematicians Got Get It Wrong. And describing how the engineers got it wrong because they didn't 358 00:32:53,990 --> 00:32:59,870 understand synchronisation. This is just one of his preliminary examples. 359 00:32:59,870 --> 00:33:05,810 So you've got the Millennium Bridge in London. This was opened in the year 2000, 360 00:33:05,810 --> 00:33:11,330 and it's only even for a couple days. Well, they had to close it. They looked at the footage. Now, 361 00:33:11,330 --> 00:33:16,520 if everyone's walking randomly, it should be fine because they're not going to match one particular 362 00:33:16,520 --> 00:33:21,560 frequency. But they looked at the footage, about 20 percent of the pedestrians. Emily, it's quite crowded at 20 363 00:33:21,560 --> 00:33:26,990 percent of them were all marching in sync. The bridge is moving about seven and a half centimetres 364 00:33:26,990 --> 00:33:32,120 either. Only 20 percent of the pedestrians, under 365 00:33:32,120 --> 00:33:37,400 the best estimation were walking in sync. Actually, if you study those videos, 366 00:33:37,400 --> 00:33:43,220 it's absolutely clear that quite a few of the heads are moving from side to side. 367 00:33:43,220 --> 00:33:48,320 I have never seen any video personally that has suggested any synchronisation in 368 00:33:48,320 --> 00:33:53,340 where they're actually placing their feet. You looked at some of those earlier videos, you 369 00:33:53,340 --> 00:33:58,470 might have seen some schoolchildren who are sort of placing their feet kind of like this as the bridge is 370 00:33:58,470 --> 00:34:04,560 moving. There's certainly no evidence that I've seen of people placing their footsteps 371 00:34:04,560 --> 00:34:10,170 in synchrony with the bridge. It's also not a good example of how against clocks 372 00:34:10,170 --> 00:34:15,480 or these cats, because the bridge has a natural frequency. The common frequency 373 00:34:15,480 --> 00:34:20,910 that these pedestrians adopted under this explanation is not some 374 00:34:20,910 --> 00:34:26,100 random frequency or some collective frequency. The average frequency of the pedestrians 375 00:34:26,100 --> 00:34:31,140 like it would be in the Kuramoto theory. But is the natural frequency of the mode of vibration 376 00:34:31,140 --> 00:34:36,190 of the bridge. And also, if that's the right explanation. What? How 377 00:34:36,190 --> 00:34:41,290 do you explain what Arop observed? And it's widely agreed that there 378 00:34:41,290 --> 00:34:46,330 needs to be a critical number of pedestrians for this to occur. Why is it when you have one 379 00:34:46,330 --> 00:34:51,370 less pedestrian than this critical number, they do not synchronise. And there isn't an instability. But when you have 380 00:34:51,370 --> 00:34:56,680 one more, it kicks off. Not understood pedestrian 381 00:34:56,680 --> 00:35:01,990 footsteps and nothing like simple sinusoidal oscillators. We walk in a very 382 00:35:01,990 --> 00:35:07,690 strange way. Essentially, we walk on flat ground with expanding, 383 00:35:07,690 --> 00:35:12,910 expending almost no energy. What we essentially do is we fall over. We balance 384 00:35:12,910 --> 00:35:18,310 on this leg and we fall over onto the next leg. And then we swing this leg through. 385 00:35:18,310 --> 00:35:23,680 And we fall over. That is nothing like a pendulum 386 00:35:23,680 --> 00:35:29,020 or a spinning generator. And 387 00:35:29,020 --> 00:35:34,060 why did an Arab negative damping explanation work and lateral instability on 388 00:35:34,060 --> 00:35:39,490 bridges? How did how do you explain instabilities? If the bridge 389 00:35:39,490 --> 00:35:44,860 isn't tuned to the pedestrian natural frequency, is it very special for this London Millennium 390 00:35:44,860 --> 00:35:50,490 Bridge, as some people claim that it just happened to that they picked 391 00:35:50,490 --> 00:35:55,990 a vibration or they picked a suspension mechanism that meant that the mode of vibration 392 00:35:55,990 --> 00:36:01,100 of the bridge was very close to a natural pedestrian frequency. Unfortunately, 393 00:36:01,100 --> 00:36:06,790 that's not the case. You also you've got. 394 00:36:06,790 --> 00:36:12,360 And one of the reasons we know that's not the case is if you dig into the literature, there'd be many examples. 395 00:36:12,360 --> 00:36:17,700 Of bridges, not just the Millennium Bridge that have gone unstable 396 00:36:17,700 --> 00:36:22,740 due to crowds walking on them, not due to traffic, but due to crowds. The Auckland Harbour 397 00:36:22,740 --> 00:36:28,980 Bridge doing a particularly marrie protest, walk a footbridge in Changi Airport in Singapore 398 00:36:28,980 --> 00:36:34,350 and then in Bristol Suspension Bridge, it was observed. So if you know anything about Bristol, 399 00:36:34,350 --> 00:36:39,770 the suspension bridge goes across a gorge which connects the city to Ashton Gate Park. 400 00:36:39,770 --> 00:36:44,880 And every summer there's a Europe's largest balloon festival in the park. And in two of those evenings, 401 00:36:44,880 --> 00:36:50,040 they have a nightclub event. And what they do is they get a massive arena and about 100 balloons get 402 00:36:50,040 --> 00:36:55,320 lit up and lots of novelty balloons get lit up as massive 403 00:36:55,320 --> 00:37:00,360 festivities cetera. And it finishes at 10 o'clock and everyone walks home. Now, the quickest way to 404 00:37:00,360 --> 00:37:05,460 walk home is across the suspension bridge. And they would close it to traffic as well so that you only get people walking 405 00:37:05,460 --> 00:37:10,740 across it. And then it was noted that the bridge started to move. So my colleague John McDonald 406 00:37:10,740 --> 00:37:17,700 was called in and the following year he instrumented the bridge. And this is what he found. 407 00:37:17,700 --> 00:37:22,800 What he found. What I'm showing you here is this is over the course of the week. And 408 00:37:22,800 --> 00:37:28,020 these are the peak lateral vibrations just after the nightclub event. 409 00:37:28,020 --> 00:37:33,110 So these are what the. They are a mess. So the 410 00:37:33,110 --> 00:37:38,210 average displacement, daytime evening, daytime evening and then the nightly event, 411 00:37:38,210 --> 00:37:43,220 massive in the lateral direction, not in the vertical or the torsion. 412 00:37:43,220 --> 00:37:48,560 And if you look in more detail, you can actually look at the record 413 00:37:48,560 --> 00:37:54,110 during the time the pedestrians were walking across. But what was noticed is that 414 00:37:54,110 --> 00:37:59,120 the as the number of pedestrians changed, there appeared to be 415 00:37:59,120 --> 00:38:04,250 a critical number beyond which added, if you can see this in the blue and the red here, 416 00:38:04,250 --> 00:38:09,260 that two different modes of the suspension bridge were excited. So 417 00:38:09,260 --> 00:38:14,360 it wasn't just it wasn't just that the pedestrians happen to be walking at 418 00:38:14,360 --> 00:38:19,550 a certain frequency, which was the mode of the bridge. The pedestrian average frequency 419 00:38:19,550 --> 00:38:24,710 was actually halfway between two different modes, one of which, if you like, is is like 420 00:38:24,710 --> 00:38:30,020 this. And the other of which was like that. And both were excited. 421 00:38:30,020 --> 00:38:35,060 How can that be synchrony? How can you get simultaneous synchronisation to 422 00:38:35,060 --> 00:38:42,340 two different vibration frequencies? There's something odd about that. 423 00:38:42,340 --> 00:38:47,620 So what about Arabs? Negative damping idea? Well, here's a really 424 00:38:47,620 --> 00:38:52,660 simple nonconstructive explanation of their idea, and 425 00:38:52,660 --> 00:38:57,700 I think they hit the nail on the head. Have you ever considered what you 426 00:38:57,700 --> 00:39:02,740 do when you're walking on ground that is moving laterally. And if 427 00:39:02,740 --> 00:39:07,750 you've never experienced this, you probably lucky there are some people in the audience. There are 428 00:39:07,750 --> 00:39:12,910 some experiments that can be done by adults that involve how can I put 429 00:39:12,910 --> 00:39:18,220 this a certain change in their perception due to circlet liquid 430 00:39:18,220 --> 00:39:23,320 intake that makes them perceive that the ground is moving. And if you've ever witnessed, don't try 431 00:39:23,320 --> 00:39:28,510 this at home, children, please. If you've ever witnessed somebody undertaking one of those experiments, they tend 432 00:39:28,510 --> 00:39:33,700 to walk in a way that is a bit sort of like this, is that they're changing their balance all the time 433 00:39:33,700 --> 00:39:38,890 as the ground seems to move away from them and doesn't seem to be very flat. 434 00:39:38,890 --> 00:39:44,140 But seriously, what are you doing? What are you doing? If you are moving, if you're on ground, 435 00:39:44,140 --> 00:39:49,180 that's moving perhaps some I don't know the theme park or something. Sometimes you step onto those rotating 436 00:39:49,180 --> 00:39:55,360 platforms. You walk on ground that's moving. What you desperately try and do is not fall over. 437 00:39:55,360 --> 00:40:00,490 And how do you not fall over? Well, if I've got some momentum in this direction. I'm going to ensure that my next 438 00:40:00,490 --> 00:40:06,110 footstep loses that momentum. I'm going to try and write myself 439 00:40:06,110 --> 00:40:11,470 to to decrease the amplitude of my oscillation. So whatever is causing 440 00:40:11,470 --> 00:40:16,660 me to oscillate, I don't care about the ground. I don't care about necessarily where I'm headed, 441 00:40:16,660 --> 00:40:21,880 although in a rough sense I might. What I care about is not falling over. And therefore I want to lose 442 00:40:21,880 --> 00:40:26,920 energy from this degree of freedom. I want to decrease their energy. Where do 443 00:40:26,920 --> 00:40:33,070 I want to put that energy? Well, there's two places I can put it. I can warm my muscles up 444 00:40:33,070 --> 00:40:38,140 or I can give it to the ground. I can push against the ground. And that's generally what we do when 445 00:40:38,140 --> 00:40:43,300 we walk. That's where we get extra momentum from when we run, etc. We do work 446 00:40:43,300 --> 00:40:48,580 against the ground. So if I want to not fall over in that direction, 447 00:40:48,580 --> 00:40:54,130 I'm going to push against the ground. And the only place that energy is going to go 448 00:40:54,130 --> 00:40:59,160 is into the ground itself. Now, normally, the ground is very stiff and it can take it, 449 00:40:59,160 --> 00:41:04,930 you know, might get impacted or whatever. But if you're on something like a suspension bridge on average, 450 00:41:04,930 --> 00:41:10,780 what you're doing is it starts to move is you're going to give energy to the bridge. 451 00:41:10,780 --> 00:41:16,060 That's irrespective of whatever speed you are walking at. And whatever 452 00:41:16,060 --> 00:41:21,100 speed the ground is moving, on average, whenever you start to wobble, you're going to 453 00:41:21,100 --> 00:41:27,570 give energy to the bridge. And that's essentially what negative dumping is. It's 454 00:41:27,570 --> 00:41:33,070 it's that the coefficient. It's the degree to which, on average, a pedestrian 455 00:41:33,070 --> 00:41:38,290 wants to give energy to the bridge. If you caused the pedestrian to move and 456 00:41:38,290 --> 00:41:43,660 therefore, if you get enough pedestrians, then they get enough of this negative 457 00:41:43,660 --> 00:41:48,670 damping to overcome the natural bridge dynamics, the natural damping of the bridge. And 458 00:41:48,670 --> 00:41:53,860 the thing will go unstable. And in fact, this is not a mysterious thing. This is completely analogous 459 00:41:53,860 --> 00:41:59,050 to a flutter instability in aerospace engineering and aerospace engineering. If you have 460 00:41:59,050 --> 00:42:04,240 the wind blow fast enough over a wing or indeed the Tacoma Narrows Bridge did something 461 00:42:04,240 --> 00:42:09,310 similar where there was enough vortex shedding that caused the natural dynamics 462 00:42:09,310 --> 00:42:14,680 to be negatively damped. And that gives rise to a flutter instability. This is also known as a hop bifurcation 463 00:42:14,680 --> 00:42:21,910 in mathematics, and it doesn't require any matching of frequencies or any synchronisation. 464 00:42:21,910 --> 00:42:28,430 But how to estimate this negative damping coefficient and this critical number of pedestrians? 465 00:42:28,430 --> 00:42:34,880 So I could show you another video, but I probably won't because I couldn't edit it very well. 466 00:42:34,880 --> 00:42:40,310 But what you can do is you can start to say, well, what actually happens when you walk? 467 00:42:40,310 --> 00:42:46,130 And basically what you need to understand is that there's a difference between the frequency 468 00:42:46,130 --> 00:42:51,230 with which you're putting your feet down and the frequency with which you're wobbling 469 00:42:51,230 --> 00:42:56,750 from side to side. Those are completely independent degrees of freedom. 470 00:42:56,750 --> 00:43:02,240 And you can have a model for telling you when you put your foot down. And sometimes the central pattern generator 471 00:43:02,240 --> 00:43:07,250 has decided I'm going to walk at this pace unless the ground stops me from doing so. But 472 00:43:07,250 --> 00:43:12,260 if I'm walking at this pace and the ground starts to move, then I might not change my pace. I might do 473 00:43:12,260 --> 00:43:17,540 a little bit, but I might adjust where I put my feet. And that's a different motion. The side 474 00:43:17,540 --> 00:43:22,730 to side motion is a different motion from the pattern generator that's 475 00:43:22,730 --> 00:43:28,340 telling me when to put my foot down. They're going to be coupled. And so these. So for simplicity, 476 00:43:28,340 --> 00:43:33,530 we could imagine that I am trying to walk at a certain speed. That's in the so-called 477 00:43:33,530 --> 00:43:38,900 sagittal plane, the four aft plane. And I'm moving 478 00:43:38,900 --> 00:43:43,940 from side to side as this lateral oscillating structure is causing me to do 479 00:43:43,940 --> 00:43:48,950 so in the so-called frontal plane. And that you can describe by various 480 00:43:48,950 --> 00:43:54,470 models of pendulum falling onto another pendulum, a 481 00:43:54,470 --> 00:43:59,610 pendulum. As I fall onto the other foot. And you can get various time 482 00:43:59,610 --> 00:44:04,670 series. So they're upside down pendulums rather than pendulums the right way up. 483 00:44:04,670 --> 00:44:09,740 And you can write down equations for these and I won't go into it. But there's a number of different equations. John 484 00:44:09,740 --> 00:44:14,810 MacDonald's original model just assumes that this is prescribed. 485 00:44:14,810 --> 00:44:19,910 So I'm walking at a certain pace and nothing will change that. Then you can look at a model that allows that 486 00:44:19,910 --> 00:44:25,760 to be adapted such that if I have to take a wider step, then I'm going to adjust 487 00:44:25,760 --> 00:44:30,890 how far forwards. And when I put the foot down. So it's it's just a geometric 488 00:44:30,890 --> 00:44:36,020 adjustment corresponding to how wide my leg is. 489 00:44:36,020 --> 00:44:42,960 And the adjustment is due to a sort of control law that says I'm going to try not to fall over. 490 00:44:42,960 --> 00:44:48,300 And then there's another model that Igor Belak and his student came up with, 491 00:44:48,300 --> 00:44:54,000 which just says, I will put my foot down whenever I feel myself going 492 00:44:54,000 --> 00:44:59,160 over the 45 degrees. Then I'll switch onto the other foot, 90 degrees, 180 degrees. As soon as I 493 00:44:59,160 --> 00:45:04,190 feel myself passing through this. I'll put my foot down. So we 494 00:45:04,190 --> 00:45:09,320 tried all three models. There are more complicated models. There are models of articulated legs. There 495 00:45:09,320 --> 00:45:14,450 are models that fully couple everything. There are models that have multiple degrees of freedom that add masses for the legs as 496 00:45:14,450 --> 00:45:19,700 well as mass for the core. There are models that contain the central pattern generator 497 00:45:19,700 --> 00:45:25,010 as part of them. But in essence, all models of walking have this notion 498 00:45:25,010 --> 00:45:30,900 of I put my weight on one foot, then I lift it and I put my weight on the other foot. 499 00:45:30,900 --> 00:45:36,030 And what we found. So we then did some mathematical analysis and the mathematical analysis 500 00:45:36,030 --> 00:45:42,270 is model independent. It says, As long as my model is like that, I've got a bridge. 501 00:45:42,270 --> 00:45:47,700 Linear oscillator with a massive mass. Just goes back and forwards depending on how it's forced. And I've got 502 00:45:47,700 --> 00:45:52,740 pedestrian's, much less mass, that provide feedback to the bridge. And 503 00:45:52,740 --> 00:45:57,840 the bridge provides feedback to them. And then what we found was that you can 504 00:45:57,840 --> 00:46:02,880 calculate from any model the degree of negative damping, and 505 00:46:02,880 --> 00:46:07,950 it contains three components. I've called them Sigma one, sigma two and Sigma three. So in 506 00:46:07,950 --> 00:46:13,200 the simple John MacDonell model, it's actually the Bosnian town model. Sigma one is the only 507 00:46:13,200 --> 00:46:18,570 thing that occurs, which is the dependence of the average 508 00:46:18,570 --> 00:46:23,640 lateral force on the bridge is velocity. So if the bridge moves a bit faster than I 509 00:46:23,640 --> 00:46:28,710 put my foot out wider. This diagram sort of explains where 510 00:46:28,710 --> 00:46:33,840 the asymmetry comes from, because you might think you might think that on average, this notion 511 00:46:33,840 --> 00:46:39,240 of me putting my foot out less wide or wider to stop myself falling over would balance. 512 00:46:39,240 --> 00:46:44,340 In other words, the number of times I push the bridge in the direction it's already going would 513 00:46:44,340 --> 00:46:49,560 be the same as the number of times I push the bridge the other way. But it isn't and it isn't 514 00:46:49,560 --> 00:46:54,690 for the following reason. If I push the bridge in the direction that it's already going, 515 00:46:54,690 --> 00:47:00,080 I put my leg down and as my leg moves. So this angle increases 516 00:47:00,080 --> 00:47:05,340 and I give a higher component of force in that direction. If I put my leg down here 517 00:47:05,340 --> 00:47:10,410 and the bridge is moving towards me. So that causes my foot to go that way and I put 518 00:47:10,410 --> 00:47:15,510 less force. My force is now going downwards. There is there is now no 519 00:47:15,510 --> 00:47:20,670 component of force in that direction. And so you see there is a fundamental difference between doing 520 00:47:20,670 --> 00:47:25,680 that well and doing that. So you see when people are walking 521 00:47:25,680 --> 00:47:30,720 on, they sometimes do this business that provides much less energy than if you do 522 00:47:30,720 --> 00:47:36,120 that and end up putting a foot out wide and pushing the bridge in the same direction. And that's where the fundamental 523 00:47:36,120 --> 00:47:41,550 asymmetry in this negative damping cups comes from. And then there's a an adjustment 524 00:47:41,550 --> 00:47:46,560 in where you put your foot. And then there's an adjustment in where you put 525 00:47:46,560 --> 00:47:51,750 your foot forwards and backwards as well. And then we run all three models and we found 526 00:47:51,750 --> 00:47:56,940 the same sort of thing. So what am I showing here? I've almost finished. So 527 00:47:56,940 --> 00:48:02,100 this is the I'm going to show you what happens as we do a simulation. Then 528 00:48:02,100 --> 00:48:08,550 add another pedestrian. Then do a simulation. Then add another pedestrian. Then do a simulation. 529 00:48:08,550 --> 00:48:14,370 So up to about 200 pedestrians in this particular model. 530 00:48:14,370 --> 00:48:19,410 We take the pedestrians to all have a similar frequency with a randomly assigned 531 00:48:19,410 --> 00:48:24,900 frequency and face with it, just a five percent variance. 532 00:48:24,900 --> 00:48:30,180 And the pedestrians, this is their amplitude here, doesn't change much 533 00:48:30,180 --> 00:48:35,250 depending on whether the bridge is stable or unstable. But what you see is the bridge 534 00:48:35,250 --> 00:48:40,320 motion here. And as soon as you pass 200 pedestrians, then the bridge motion takes 535 00:48:40,320 --> 00:48:45,430 off. And in this model, there is nothing to stop it. Now, two things you notice. 536 00:48:45,430 --> 00:48:50,860 One is the negative damping coefficient, the damping curve fish out of 537 00:48:50,860 --> 00:48:55,990 all the pedestrians minus that of the bridge. 538 00:48:55,990 --> 00:49:01,360 It changes sign at this 200 and the amplitude starts to grow. And 539 00:49:01,360 --> 00:49:06,850 this here are is the Kuramoto order parameter. Remember, that was that was the degree 540 00:49:06,850 --> 00:49:12,400 of how much the oscillators are in sync with each other. And in this case, there's no difference 541 00:49:12,400 --> 00:49:17,530 between after the bridge amplitude. That's this here grows 542 00:49:17,530 --> 00:49:22,960 very large. And was the bridge amplitude is small. Then we added this feedback 543 00:49:22,960 --> 00:49:28,030 to the model. And the feedback is I change how often I put my foot or 544 00:49:28,030 --> 00:49:33,040 I change how far forwards I put my foot, depending on how wide I am putting my feet. 545 00:49:33,040 --> 00:49:38,470 If you include that feedback, then something really interesting happens. You do see a large degree 546 00:49:38,470 --> 00:49:44,710 of coherence at least, or a change in the order parameter apparent synchronisation. 547 00:49:44,710 --> 00:49:50,080 This is, if you like, 20 percent of the pedestrians appear to be walking in step. 548 00:49:50,080 --> 00:49:55,210 But notice, this happens as a result of the instability. The damping coefficient 549 00:49:55,210 --> 00:50:00,850 goes negative. The amplitude increases. And then as the amplitude increases, 550 00:50:00,850 --> 00:50:06,850 that causes more of this. What's sometimes called phase pulling or coherence 551 00:50:06,850 --> 00:50:12,880 amongst the pedestrians. You can also plot the 552 00:50:12,880 --> 00:50:18,160 here I am plotting the bridge frequency dividing by the pedestrian frequency. 553 00:50:18,160 --> 00:50:23,200 And here I am plotting the damping coefficient. And what you find 554 00:50:23,200 --> 00:50:28,480 is that this damping coefficient can actually sometimes. This is the amount of damping per 555 00:50:28,480 --> 00:50:33,820 pedestrian or the amount of negative damping per pedestrian that actually its frequency 556 00:50:33,820 --> 00:50:38,860 dependent. It's a question of timing. What I was saying about putting your foot out wide and 557 00:50:38,860 --> 00:50:44,530 putting your foot out was kind of assuming that I am walking at a similar pace to the bridge's oscillating. 558 00:50:44,530 --> 00:50:49,660 If the bridge is oscillating much faster or much slower than actually during the time 559 00:50:49,660 --> 00:50:54,680 that I'm on my right foot, the bridge might have moved back. And so there's an element of timing. 560 00:50:54,680 --> 00:51:00,130 So it's actually it is frequency dependent. But what we find here is that between a frequency 561 00:51:00,130 --> 00:51:05,530 ratio of point eight and two of the bridge frequency to the pedestrian frequency, the bridge is liable 562 00:51:05,530 --> 00:51:10,640 to this instability. Because these curves are positive, these three 563 00:51:10,640 --> 00:51:16,160 different curves are for different pedestrian frequencies. So I'm plotting this just as a ratio. 564 00:51:16,160 --> 00:51:21,200 It isn't actually only a function of this ratio. It depends on the underlying pedestrian or bridge 565 00:51:21,200 --> 00:51:26,960 frequency. The blue curve is an analytic curve, which you can derive from the simple model. 566 00:51:26,960 --> 00:51:32,060 And this here is in simulations, the number of pedestrians it took 567 00:51:32,060 --> 00:51:37,250 to make the bridge go unstable. So if there's this large frequency mismatch, you can never 568 00:51:37,250 --> 00:51:42,290 make it go unstable. These are violin plots that show you a big scatter because there's a scatter 569 00:51:42,290 --> 00:51:47,480 depending on all kinds of variations that are here. But essentially, you do not have 570 00:51:47,480 --> 00:51:52,820 to be in this regime where there's a one to one ratio 571 00:51:52,820 --> 00:51:58,010 between the bridge and the pedestrian frequency. And in fact, that's not optimal. About one point three or one point four 572 00:51:58,010 --> 00:52:03,230 ratio were optimal in the sense of trying to trigger the instability. The worst case seems 573 00:52:03,230 --> 00:52:08,300 to be around. One point four is the ratio between the bridge frequency and the pedestrian frequency. Then 574 00:52:08,300 --> 00:52:13,310 it only takes about 150 pedestrians for these parameters. 575 00:52:13,310 --> 00:52:18,500 And we could do the same for the falling over model, the Belic et al model 576 00:52:18,500 --> 00:52:23,570 that I mentioned. And this one also shows a very large amount of coherence. 577 00:52:23,570 --> 00:52:29,300 This one does synchronise and already been used as a as an example of synchronisation. But once again, 578 00:52:29,300 --> 00:52:34,490 synchronisation happens after the instability, the instability, whether negative damping first kicks 579 00:52:34,490 --> 00:52:39,590 in, occurs at 156 pedestrians. And then at least for these simulations where 580 00:52:39,590 --> 00:52:44,960 we're adding the pedestrian slowly. It took a little bit later before the amplitude saturated and the amplitude 581 00:52:44,960 --> 00:52:50,180 saturates as the pedestrians now all start to fall into sync. You see, 582 00:52:50,180 --> 00:52:55,310 prior to the instability, they all have random phases in plotting a sample, 583 00:52:55,310 --> 00:53:00,350 at least of the pedestrians as sine waves. But here they're most of their sine waves or lie 584 00:53:00,350 --> 00:53:05,510 on top of each other. And we can do a similar thing and find sorry, 585 00:53:05,510 --> 00:53:10,520 something's got chopped off here. This damping coefficient in the number of pedestrians. So 586 00:53:10,520 --> 00:53:16,820 to sleights almost there direct conclusions. Partial synchronisation 587 00:53:16,820 --> 00:53:21,860 sync this 20 percent of pedestrians might synchronise, probably is 588 00:53:21,860 --> 00:53:27,080 a property of pedestrians on laterally excited bridges, on laterally moving bridges. So probably 589 00:53:27,080 --> 00:53:32,210 is a thing. It's not a bad theory, but I claim it doesn't 590 00:53:32,210 --> 00:53:37,550 explain the instability. Doesn't have explanatory power. It's an observation 591 00:53:37,550 --> 00:53:42,890 of what happens. Doesn't explain why it happens. And in particular, pedestrians are not like Horgan's 592 00:53:42,890 --> 00:53:48,050 clocks. And the bridge is not a passive coupling device. It has its own 593 00:53:48,050 --> 00:53:54,170 natural frequency. It's not like the pedestrians are choosing to synchronise on their 594 00:53:54,170 --> 00:53:59,240 average frequency. And it requires, therefore, the pedestrian walking frequencies to be 595 00:53:59,240 --> 00:54:04,880 tuned to the frequency of the bridge. And it also gives no insight 596 00:54:04,880 --> 00:54:10,220 to the engineers on how to measure the critical number of pedestrians. In order to 597 00:54:10,220 --> 00:54:15,250 have a safe bridge. This concept to negative damping, I claim, 598 00:54:15,250 --> 00:54:20,350 is intuitive. It's basically what I do in order to not fall over when 599 00:54:20,350 --> 00:54:25,810 I'm walking on moving ground. What I'm doing is I'm giving energy 600 00:54:25,810 --> 00:54:32,140 to the bridge to stop myself having too much energy in the lateral direction. 601 00:54:32,140 --> 00:54:37,210 This effect can be measured. You can measure Arop measured it. You can model it. And 602 00:54:37,210 --> 00:54:42,730 you can use it as part of a simulation. It's a robust it's robust to the choice of mathematical 603 00:54:42,730 --> 00:54:47,770 model. We took three different mathematical models and they all showed this negative damping 604 00:54:47,770 --> 00:54:53,650 and the negative damping showed when the numerical simulations go unstable. 605 00:54:53,650 --> 00:54:59,440 It explains instability across a wide range of Brigg fruit bridge frequencies. It's not just tuned 606 00:54:59,440 --> 00:55:04,510 to the idea that the pedestrians have to be walking at the bridge frequency or indeed that they have to be walking at 607 00:55:04,510 --> 00:55:10,480 similar frequencies to each other. And it gives the engineer a way to 608 00:55:10,480 --> 00:55:15,490 compute this critical number. Protests. Pedestrians. And there's a nor 609 00:55:15,490 --> 00:55:21,370 there, we should say. And the necessary bridge damping to avoid instability. 610 00:55:21,370 --> 00:55:27,240 So in a sense, in a sense, the engineers were right all along. 611 00:55:27,240 --> 00:55:32,640 All this hoo ha! All this stuff about saying, wow, what a great example of synchronisation. 612 00:55:32,640 --> 00:55:37,770 It probably isn't. There was probably some synchrony, but actually a large 613 00:55:37,770 --> 00:55:42,900 amplitude. There always is some synchrony. But it was probably more like these cats that 614 00:55:42,900 --> 00:55:48,000 I have here in that there was precious little synchrony going on. 615 00:55:48,000 --> 00:55:53,040 But that's not the point. The point is, it's this negative damping 616 00:55:53,040 --> 00:55:58,410 criterion, which is something you can calculate is nowhere near as famous. It has 30 citations, 617 00:55:58,410 --> 00:56:03,870 whereas the nature paper on synchronisation has several hundred, maybe a thousand. I'm not sure 618 00:56:03,870 --> 00:56:09,570 it's been there in the engineering literature for a long time. It's just an inconvenient truth. It's 619 00:56:09,570 --> 00:56:14,820 it's if you like, if I'm can be a bit more philosophical. The synchronisation of near identical oscillators 620 00:56:14,820 --> 00:56:20,920 is a great mathematical theory that explains all kinds of things, but it doesn't explain everything. 621 00:56:20,920 --> 00:56:26,130 And in my view, it doesn't explain why and it doesn't predict when 622 00:56:26,130 --> 00:56:31,260 bridges go unstable to lateral oscillations. So maybe maybe this 623 00:56:31,260 --> 00:56:36,690 is a case of what Thomas Huxley described as the great tragedy of science, the slaying 624 00:56:36,690 --> 00:56:41,760 of a beautiful hypothesis by an ugly fact. But maybe 625 00:56:41,760 --> 00:56:47,030 not. Maybe actually, it's a failure to apply 626 00:56:47,030 --> 00:56:52,090 the core principle of mathematical modelling. Which is Ockham's Razor. If 627 00:56:52,090 --> 00:56:57,460 there exist two explanations for an occurrence, the one requires the smallest number of assumptions 628 00:56:57,460 --> 00:57:02,550 is usually correct. Sometimes we go out there with 629 00:57:02,550 --> 00:57:08,010 our hammers as mathematicians looking for nails and saying, oh, I am an expert in synchrony, oh, look, there's a perfect 630 00:57:08,010 --> 00:57:13,140 example of synchrony without actually checking the assumptions, without saying what is 631 00:57:13,140 --> 00:57:18,220 truly oscillating here, what a truly the natural frequencies, et cetera. Or, if 632 00:57:18,220 --> 00:57:23,230 you like, keep it simple, stupid. If I've learnt anything, is a mathematical 633 00:57:23,230 --> 00:57:28,320 modeller. I almost never. Make 634 00:57:28,320 --> 00:57:33,800 progress or have insights by making something more complicated. It's nearly 635 00:57:33,800 --> 00:57:38,900 always the case that the truth is much simpler than you think. You just haven't been 636 00:57:38,900 --> 00:57:44,050 looking at it the right way. And avoid the wisdom 637 00:57:44,050 --> 00:57:49,240 of the crowd, popular explanations are often wrong. There's a there's a physics textbook 638 00:57:49,240 --> 00:57:54,610 explanation. There's another bridge instability, very famous in physics and engineering textbooks and calculus 639 00:57:54,610 --> 00:57:59,800 textbooks that says this is an example of resonance where the wind started resonating 640 00:57:59,800 --> 00:58:05,080 or the vortices shed from the cables of a bridge resonated with the bridge's natural frequency. 641 00:58:05,080 --> 00:58:10,300 Balderdash, or some of the popular explanations you see of how aeroplanes generate 642 00:58:10,300 --> 00:58:15,700 lift or even explanations I've seen for why there is a Nobel prise for economics. 643 00:58:15,700 --> 00:58:21,190 There isn't. And why there was never one for maths, which has nothing to do with 644 00:58:21,190 --> 00:58:27,370 Nobel and Mittagong Lefler and their personal relationship or otherwise. 645 00:58:27,370 --> 00:58:32,440 So finally then, if I can give you some messages, it's think. 646 00:58:32,440 --> 00:58:37,700 Never be afraid to think for yourself. Be a little bit scientifically stubborn. 647 00:58:37,700 --> 00:58:42,910 Don't follow the wisdom of the crowd. Don't say just because everybody else says it's an example of one of these. Find 648 00:58:42,910 --> 00:58:48,730 out for yourself. Question. Cheque the assumptions. Apply Ockham's Razor. 649 00:58:48,730 --> 00:58:54,040 What actually are the hidden assumptions. Am I doing here? Am I making this problem more complicated 650 00:58:54,040 --> 00:58:59,370 than I need to? And sometimes sometimes we do that. And I think mathematics 651 00:58:59,370 --> 00:59:04,500 is a very good tool for cutting out assumptions and making things simpler if we only 652 00:59:04,500 --> 00:59:09,540 did it the right way. And so finally, then, I'd like you to remember that rhythmic clapping without 653 00:59:09,540 --> 00:59:37,040 being presumptuous. And thank you very much.