Welcome to all of you watching streaming or catch up. It's a great pleasure. I would say to be back
in Oxford where I started, of course, it was in a different building in those days. But I have been in this building
a number of times before, and it's a great honour and privilege to give this lecture today. So I
slightly change the title. I wanted to call it on the Millennium Bridge Synchronisation Myth.
And the subtitle is Why Pedestrian Bridges Wobble Synchronisation and the
Wisdom of the Crowd and Hear. The Wisdom of the Crowd refers to how crowds behave on
moving structures and how crowds behave generally, but also how the wisdom of the crowd
can sometimes, in my view, lead to scientific theories and something that inverted
commas. Everybody knows, don't they? That might not be true or might not be as simple as
that. And I should say at the outset, this is work with my colleague John McDonald in civil
engineering and his former APHC student, Matt Bostian, who's now at the University of
Leicester, and also Igoe Belic, Russell Jita and Kevin Dailey, who were in Georgia
State University. So you may have seen, I don't know, 20
years ago. It is now almost 20 years ago, you may have seen videos
or you may have been present at what happened on so-called London Millennium Bridge. Now,
the London Millennium Bridge was billed as the world's first horizontally sprint
suspended suspension bridge. And it was. It links the Tate
Modern Gallery was in Paul's Cathedral. It's an iconic structure. And it was billed
as it won awards because of its unique heritage or its unique design
as a collaboration between an architect and an engineer. But on the day
it opened, this happened. The Millennium Bridge is a stunning aluminium and steel structure.
A bit of this is a BBC programme. All of my videos in Italian, it's night and
day, 10th of June 2000. But as soon as
the crowds hit the bridge, there was a problem. It wobbled and not just a bit, but enough to
make people clinging to the handrails for security. Clearly, the crowds themselves
were driving the bridge in a completely unexpected and very worrying way.
The bridge was closed three days later and it's remained closed ever since. But how could this have happened?
Why wasn't it? How did it happened? Well, I want to go back into the said the popular theory.
Roughly speaking, is when you look at that video, I don't know what you saw, but you might
have seen a whole lot of heads that appeared to be moving in synchrony. And
there is a strong belief that what happened was that as people started to synchronised their footsteps.
So the bridge started to have its natural frequency amplified and as its
frequency was amplified, so that synchrony increased. And this
is used as almost a I would say, a sort of a billboard poster or a canonical example
of the mathematical theory of synchronisation. It was very convenient because the year 2000 was when that
theory was really taking off. So I want to tell you about the mathematical theory of synchronisation.
And it goes back to Christian Horgan's who until did Cobb Parkway Station.
I didn't realise had two A's in his Christian name. But as always, I'd been preparing these
slides right at the last minute and so called Hagans clocks.
Then there are lots of other examples of synchrony. And you if what I'm saying
is boring or not interesting, you can look at these cats and decide whether they are synchronising
or not. Also, brain waves, power grids are power grids rely on
the idea that the whole of the AC circuit that supplies power to the whole
of a single grid like the nation is completely in synchrony. So, you know, that AC circuit
is essentially well, you can think of it as a wave vector of polarised
electromagnetic radiation, but that has to be completely in sync across the whole network. Otherwise
it goes down and rhythmic clapping. Well, how does that come about? We'll talk about that
then. I want to talk about Yoshiki Kuramoto, who came up with a really simple
mathematical theory of synchronisation, which I will show you. At least I will do a bit of maths on the board
and then give you a hint. How you can go from two to many oscillations
and then give you the explanation for them under a millennium bridge. But really, this is going to be a tale
of two bridges, because there's another theory of how
the London Millennium Bridge and other footbridges might go unstable. And I might call that theory.
How not to fall over. Quite a simple theory. It's a bit like Douglas Adams, How to Fly,
which is aim for the ground and then miss.
And this was actually triggered, amongst other things, by my colleague John MacDonald's observation on another iconic
bridge, the Bristol Clifton Suspension Bridge. And I'll explain to you his model
of walking and the idea of so-called negative damping, averagely speaking
and then simulating and some results from some three simple models
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
is no such thing as a perfect answer. When you doing mass of the real world. And I might
give some philosophical remarks or I might run out of time. I certainly
will run out of time because as I was telling Diarrhoeal earlier, who runs these? My general
way that I give lectures and, you know, I get student feedback that says enthusiastic.
Yes. Five out of five. Did we understand it all? Can we follow it for the exam? Yeah,
you did. What was the best thing about Professor Champion is lecture the first time he told the joke.
What was the worst thing about his lecture? The sixth time he told the joke. So I tend to
prepare meticulously and then say the first thing that comes into my head. So I'll almost certainly
go over time, for which apologies, but I will speed up.
So Christian Horgan's was a sort of contemporary of Newton, and he's a physicist,
a mathematician, but also an inventor. And he described it discovered by chance
an unexpected or odd sympathy that he observed at home. And
here's a diagram of what he had. He had two pendulum clocks suspended, but on a
rack between two shelves. And people have done modern versions of this.
So these are not pendulum clocks. These are metronomes. They're not quite tuned to the same
frequency and they're not in sync. And what we're now going to do is we're gonna couple them
or this physicist is going to couple them. This YouTube is going to couple them by
putting them on a plank, rather like I have there, and watch what happens or listen
to what happens.
Just a tiny coupling between the pendulum's by the fact that
they are all uniformly contributing to the same sway of this plank,
that they're all coupled together through the vibration of the plank causes them
all to fall into sync. And the plank to start to move.
Oh, no, I don't want to do that. Thank you. And in fact, this idea of synchrony
has been observed in many other situations in the
physical world. So this is a picture this isn't a video because I couldn't find it of
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.
And there was the conferences in Beijing and there's the obligatory afternoon expedition
to the Great War. But it tipped them frankly, it was absolutely pouring. I'm doing my
best not to swear. I very nearly said a rude word, but it was it was raining cats and dogs. Fortunately,
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
who became fascinated by this display of waving cats. And you might be able
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
that if you look at each row is in sync with each other. This row here is
in there all in sync with each other. This row here is approximately in sync. And this row here
is approximately in-sync. And we started playing games. If what if we take one of the cats out? What if
we move one of the cats so that it moves its arm? What if we swap and turn them backwards, etc.?
Great fun for mathematicians and physicists, probably of no relevance whatsoever. These are
fireflies.
And after a while, looks towards the end of that clip. I hope you can see
that they're beginning to synchronise. Todd, find yeah. Here.
After a while, when there's a sufficient density of them, they start to synchronise. This
is what happens in Powergrid. So this is an example of two electrical generators
that are being made, being driven at slightly different speeds, slightly different voltages,
and then an electrical coupling between them is switched on. And watch what happens. So these are
two different electrical generators.
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
not the whole room. It's just the video.
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
circuit and being part of the same circuit is going to be enough
when he switches on that coupling. Now,
he hasn't changed the drive to either machine, but they synchronise to the common rotation
frequency and the common face. It's just a weak
electrical coupling between them. Whoops.
We go back and this is a video of an audience clapping.
How does that happen? Was there somebody and the conductor there wasn't on the stage going?
Now, now, now, now. So not only did they all hit the same
frequency, but they all hit the same frequency and the same face. In fact, there has been
a very interesting study recently of somebody looking at this. And they discovered one of the things that
you might notice about the two halves of that clip. I'll show it again. It's hard to see
this because you're seeing the collective clapping in the beginning, but actually it's hard
to synchronise your clapping if you clap too quickly. Generally, clapping is liable
to be synchronised if you stop to clap more slowly. And they discovered
in communist countries that when the there was a fantastic
speech that everyone enjoyed. There was a lot of clapping when the Dear Leader gave a speech
that they were obliged to clap. Suddenly there were synchrony. Now you draw your own conclusions
about why that might be. But listen to this again. Just watch an individual. They're clapping quite quickly.
Quite enthusiastically. Slightly less enthusiastic. And it slows
down into synchrony. That's interesting. I don't quite know why. So what
am I going to talk about? Well, as I was sort of intimating there, there's two ways you can make
independent oscillatory things oscillate. There's the orchestra kind of way. Orchestras
are in sync because there's a conductor, there's a conductor who says now everybody make the
same down string on your bow or for the violinist. There's a leader of the orchestra. There's a follow my
leader. But there's also a spontaneous synchronisation, like much like
you might get an ensemble piece. And I'm gonna talk about this spontaneous kind
of synchronisation because there was certainly nobody on the Millennium Bridge. You do get synchronisation, by
the way, when you play music. Another way of getting synchronised clapping is during
rock concerts. And in fact, there can be problems in when rock concerts are played
in sports stadia because sports stadia are designed to take a certain weight of
crowd. What they're not designed to take is a certain weight of crowd who are all doing this
at the same time in the stands. And there've been examples where huge instabilities have been
set up due to orchestrated synchronous jumping.
So the spontaneous case of the fireflies. There's nobody telling the fireflies when to turn off. There's
nobody telling you when to clap. The metronomes. There was nothing to tell the metronome which
face to choose. And these even cats that may or may not be synchronising.
There's nothing to tell them what face to take. And there's a fundamental theory
due to Kuramoto of phase coupled oscillators. And his very clever
idea is consider n. Mathematicians like to call things
and or Upsilon or whatever. So N is some large number, large being two or more.
That's a private joke. But then imagine that they're coupled in
some way. Perhaps they're too all coupled but weakly. And I'm not going to say what week means. I'm gonna deliberately
say I'm not going to say what coupling is. But they're all things that want to oscillate and they want to oscillate.
And I don't care how they oscillate, whether it's backwards and forwards or whether it's round and
round. But I'm going to describe their oscillation by its phase. In other
words, it's fraction of a period. So I'm not I'm not I'm going to throw away all the
physics of simple harmonic motion or whatever, and I'm going to suppose that these are things that are naturally
oscillating. So I can represent that motion in some sense as a circle of phase.
I go from zero round to 360 degrees, et cetera.
And so how we might write that mathematically is I describe this phase variable Phi
or Phi I and defy D.T. Newton invented
this dot notation for rate of change. That means the rate of change is just
Omega. What's Omega? Omega is the frequency. So in other words, I'm just something that is constantly changing
my face at a constant feed speed omega and then going back
again and starting again. So I'm going round and round, if you like, at a constant speed.
And then I'm going to suppose that I'm listening. In some sense, and I don't know how that listening is working,
it may be listening through my ears. It may be listening through the vibrations in the plank. It may be magnetic with those
cats, I'm not sure. It may be through current and voltage, but it's kind of weak.
There's some weakness that says if my neighbour or something that I'm coupled
to has a different phase to me, I'm going to adjust. There's some feedback
that says if we're not in the same phase, I'm going to sense that I don't know what that feedback
is. It's some function. I'm just going to call F and there's some coupling strength kapper,
which is this weakness. So I'm I'm. This if you look at these cats, the phase
is how far through the cycle of backwards and forwards I am. And Cappa is the extent to which
I am being influenced by one of my neighbouring cats or one of the further away cats.
So these are the small couplings and they're typically very much less than one.
And if you think about it hard enough, you realise this only makes sense in this phase coupled sense,
if the function F is symmetric, if it's to pay periodic, and
I assume that if there's no phase difference, then these things are just happy to oscillate.
So F of zero is zero and they realise the simplest functions that do that
are just simple sine waves. So the simplest thing you can take if
you know about how to decompose functions that or signals that aperiodic
into sine waves, you would say you take a one mode Fourier truncation. You just you
just take the simplest possible function. You just take the simplest possible law. I don't care what it is. I'm just
going to take the first term in some expansion. And that turns out to be a
sine wave. What is the sine wave sign in here is just a
sine wave is just something that does this. This is a sign of PHI
from zero up to 360 degrees.
As a function of PHY, so I'm just going to assume that I have these weekly coupled oscillators
and in the case of all tool coupling where the coupling strength is the same, then Kuramoto
first wrote down these equations and you can do really interesting things with these equations. What you find, I'm
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
N is two. OK. So I'm gonna take two oscillators.
And I'm going to assume that this kapper is just some constant and on the previous slide.
I had a cappa over end just to give you some scaling, but that's not strictly necessary. So, in fact,
this kapper is kapper over two from the previous slide. Woops, sorry. In fact, it's two cappa.
No, it isn't, it's kapper of it. Oh, sorry. Alan, shut up. Sorry, I didn't say that aloud, did I? No,
glad. And let Delta Omega be the difference between
these two frequencies. So these are these two generators that are going round fires is the angle of
that generator and they're going round it very slightly different speeds.
And now let me take the difference. Let me take the difference. And I'm just going to subtract these two
equations if I just subtract these two equations. I get the following.
After a tiny bit of algebra, I find that the difference between them,
the psi dot. So that's five one minus five to. I subtract
these two equations. I find this side dot is just the difference in the Omega's
minus two times Kappa PSI. Then for this equation, I can prove some things.
I can prove to you that if the coupling strength is large enough.
And the frequency difference is small enough. Then this equation
has two different equilibria. That is two different. Constant
face solutions where both the face different sorry, the constant face, different
solution, where the two oscillators will be in phase. And this is really easy to show because let me look at that equation.
What is it? What does it say? It says fi dot, which is a right.
Five dot is the rate of change of fi.
As a function of FYE. Is.
Delta Omega, let me assume Delta Omega is positive, it doesn't have to be. So that's the difference in these
two frequencies. That's fairly small. Minus
something to Kapper Times Sign of fire will sign is a graph that I
had previously. So that's gonna be a function that looks something like this.
And there's my 360 degrees. And there's my zero degrees. And that's badly
drawn because this here should be at
two kapper. Careful. Poor
minus. Delta Omega.
What have I done wrong? This is totally rubbish. This should go up like this. That should be
a zero point. There should be a zero point and I should come back to there, and that should be my 360
degrees. But notice what this does. If this.
To Kapper minus Delta Omega, as I've drawn it, is
less than one. If that down there is less than one, then I have to zero crossings.
What happens at these zero crossings? Well, this graph tells me everything, because this
if this is my angle, phy, this graph tells me whether five varies positively.
Or five areas negatively, so here, defy duty is positive, which means if I start here, I go this way,
I increase my face difference. But if I start if in this region.
Defied E.T. The rate of change of FI or 5.0 is negative. So I go this way
and I end up here. And similarly, if I start in this region
because defied fighting is positive, I go this way and I'm going to
end up here. And I go right round 360 degrees become zero
degrees again and I'll end up here. But see what that does that shows me not
only are there two zero crossings, fixed points, points of
which fire doesn't change the rate of change of fire, zero. It shows me that this one
is stable. Because if I start slightly to the right of it, I pop back,
if I start slightly to the left, I pop back. And this one is unstable.
If I start this side of it, I go all the way round the world and come back to here. And if I start
this side of it, I come back to here. That's dynamical systems one oh one. We have done
a stability analysis of a dynamical system. Just by looking at whether something is positive
or negative. And that shows me that there is a state and we call this the synchronous
state where the difference in fire is small. It's close to zero
and it's constant. And this provided this Delta Omega divided
by two K is less than one. If everything is positive, so provided
if this condition here is true. Then
I'm going to send tend to this in phase equilibrium rather than the outer phase one,
and that's going to be stable. Now, you can generalise that analysis. That was an analysis with
just two oscillators. If I have many for simplicity, let's suppose everybody
interacts with everybody else. Then you can do some things statistically. You can write down what's called
a master equation. You can assume that the whole ensemble
of all of these things, on average, you can add together how they oscillate.
And you can say that. You can say that if you look at how something oscillates and you look at its X
component. So if I think of something that's going round in a circle and I look at it from the side,
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
like a cosine wave, which is the same as sine, but just shifted by 90 degrees.
And that tells me that I can define if I average everything. Sorry, this
don't worry about this, I. That just means I'm looking from the top or from the side effectively.
But there's a neat way of writing that which is called Euler's formula that we don't need to worry about. But
essentially I can do some averaging and say on average, I can think of this
as a big oscillator. Multiplied by
something that's going round a circle, and depending on whether that average
is depending on whether that if the average is one, that
tells me that everything. Is in synchrony if that average
is zero. It tells me that they're all over the place. There is
no there is no synchrony at all. And this is called an order parameter, and Kuramoto invented
this order parameter that says how much of this is in synchrony
and how much is it not in synchrony? And more recently, people have been interested in what happens
in the boundary between synchrony and non synchrony. In the case where the coupling strength is
not really large enough to cause complete synchrony. But it's not zero. There is some coupling.
And these are being called Chimaera States. chimaeras are supposed to be creatures that are composed of two different
organisms, like a lion with a with a donkey's
head or something. And the idea is you can have these oscillator, some of which want to synchronise
and some don't. So here are some simulations, again, just
pilfered off the web of the case when the coupling strength is
large enough and the case where it isn't. So in this simulation on the left, there's just
a whole load of coupled oscillators with representing their phase, the degree of half hour round they've
gone. And in this case, it's for purely illustrative reasons. They're
not all represented on a circle, but so that you can see them all there draw. They're pulled out
transversely to the circle. So some of them are further away and some of them are nera in. That's purely for illustration.
So if you take some that a starting and random position
as you allow them to oscillate. So they start to swarm together
and very quickly all start to oscillate with the same face. So this would
be a case where the coupling strength divided by the phase
difference is large enough to cause all these oscillators to fall into sync.
And here is a case where that's not the oh, sorry about the music.
There's a case and you can watch this for hours where the coupling strength isn't quite large enough, but
you see this patch of fairly synchronous behaviour. These these oscillators are trying
approximately to follow each other. Now, I told you that when I was at the Great Wall
of China, myself and some other delegates at this conference found these
waving cats that were perfectly in synchrony. Unfortunately, this weekend
when I went and bought these cats and thought, yeah, I know what I'll do, I'll put them on the. Oh.
But then I realised what these guys are doing is really interesting. They're not synchronising,
but they are interacting with each other. What seems to happen if we look at this guy in the middle? Guy
being a gender neutral term. He speeds up and slows down as large
amplitude. And then as he slows down, the next one gains a large amplitude. As he slows down, the next
one does bounces back. Then this one's got large amplitude. Then this one's got large amplitude.
Then this one's got large amplitude. There's a travelling wave going on. And this is one
of the things oscillators can do. If they don't have enough coupling strengths to cause them
to fall into synchrony, you'll get this travelling wave of behaviour of large and small amplitude.
And that seems to be what's happening in the experiment that I tried to do.
So. What does this got to do with bridges?
Well, the accepted, if you like, the textbook example of
synchronisation and the Millennium Bridge, the explanation goes as follows.
You think of the bridge as having a lot of pedestrians in each of the pedestrians are approximately trying to
walk at the same frequency, approximately. But as the bridge just
slightly moves, just those slight movements of the bridge is enough to push them more towards
synchrony. And that's synchrony is enough to then start to make the bridge move. That then
is enough to start to couple everything together. And that's essentially what this video shows with
metronomes. If I take a enough metronomes and put them on this again, nicked off the Internet
and take a large bridge like structure eventually.
Almost like soldiers moving with military precision. The fact
that they're all couples through the collective motion of the bridge is enough to make them slowly
fall into approximate synchronisation. Left, right, left,
right. Left, right, left. Right. Company. Halt,
etc. So the full explanation
goes something like this. This was the world's first horizontally suspended bridge. Bridges are designed
generally to be vertically suspended. Suspension bridges have cables that come from above. And a lot
of effort goes into making sure that there aren't resonant frequencies in the
vertical vibrations. And, for example, it's been known for a long time. There's a command
amongst troops to break step when walking across a bridge. Because if troops were to march
across a bridge all in sync with each other, it was known that that caused large vertical
vibrations of the bridge. So it was believed that as long as it wasn't soldiers marching,
that pedestrians, all of whom have random phases and walk randomly across a bridge,
were never likely to cause a bridge to go unstable. And so it seems they
don't in the vertical domain. But they appear to in the horizontal domain.
So if you take a bridge that has a horizontal elasticity and this one had a very large one
because it was horizontally suspended approximately, then this
synchronisation of footsteps, given sufficient numbers, was enough to cause
synchrony and therefore an instability. And it doesn't really matter. So this theory goes how
you model the pedestrians because they're all oscillators. And Kuramoto theory just relies
on face coupling. It doesn't rely on the particular details of the oscillators. And it looks a bit
like having a lot of Horgan's clocks or Hagans metronomes. And this is the parodic
I can never say that word. It's a paradigm example of synchronisation. For example,
have a look at the book Sync. It's an excellent book. In Steenstra, Getz's
book in Penguin. And only about a month ago, I heard Hanna Fry, who is also superb
on six music in the Morning, talking about synchronisation and using the London Millennium Bridge example
as a classic example of synchronisation in the real world. Ashley,
the engine is Arop, were called into reengineer the bridge. The bridge was closed for two years.
And what they did is they made empirical observations. They didn't understand all this theory of synchronisation.
They just made empirical observations. And they noticed that on average,
a pedestrian provides some lateral force and that force appears
as damping. But it appears as damping with a negative sign. So they they they
said we can think of each pedestrian as like a negative damper, whatever that means. And they drew
a straight line that actually got their employees and volunteers to walk across the bridge when it was closed
and took all this data and drew a straight line through it and found that there is a a
negative damping coefficient, whatever that means. And what they concluded was the way to solve this problem
is stick ugly whacking great dampers on the bridge. Stupid engineers didn't understand
the theory. That's kind of not what anyone's ever said. But what mathematicians and physicists sometimes
think. I can remember being told as an undergrad and a maths degree. Come on. Think you're
thinking like an engineer. As if that was kind of empirical and not deep enough.
But. If you look at this was also from last year. Another fantastic
lecturer, Matt Parker. He describes himself as a stand up mathematician. If you've ever seen him
or you'll know what I mean. But if you haven't, I strongly recommend seeing one of Matt Parker's lectures. This
is just last year in the Royal Institution and a lecture entitled When
Mathematicians Got Get It Wrong. And describing how the engineers got it wrong because they didn't
understand synchronisation. This is just one of his preliminary examples.
So you've got the Millennium Bridge in London. This was opened in the year 2000,
and it's only even for a couple days. Well, they had to close it. They looked at the footage. Now,
if everyone's walking randomly, it should be fine because they're not going to match one particular
frequency. But they looked at the footage, about 20 percent of the pedestrians. Emily, it's quite crowded at 20
percent of them were all marching in sync. The bridge is moving about seven and a half centimetres
either. Only 20 percent of the pedestrians, under
the best estimation were walking in sync. Actually, if you study those videos,
it's absolutely clear that quite a few of the heads are moving from side to side.
I have never seen any video personally that has suggested any synchronisation in
where they're actually placing their feet. You looked at some of those earlier videos, you
might have seen some schoolchildren who are sort of placing their feet kind of like this as the bridge is
moving. There's certainly no evidence that I've seen of people placing their footsteps
in synchrony with the bridge. It's also not a good example of how against clocks
or these cats, because the bridge has a natural frequency. The common frequency
that these pedestrians adopted under this explanation is not some
random frequency or some collective frequency. The average frequency of the pedestrians
like it would be in the Kuramoto theory. But is the natural frequency of the mode of vibration
of the bridge. And also, if that's the right explanation. What? How
do you explain what Arop observed? And it's widely agreed that there
needs to be a critical number of pedestrians for this to occur. Why is it when you have one
less pedestrian than this critical number, they do not synchronise. And there isn't an instability. But when you have
one more, it kicks off. Not understood pedestrian
footsteps and nothing like simple sinusoidal oscillators. We walk in a very
strange way. Essentially, we walk on flat ground with expanding,
expending almost no energy. What we essentially do is we fall over. We balance
on this leg and we fall over onto the next leg. And then we swing this leg through.
And we fall over. That is nothing like a pendulum
or a spinning generator. And
why did an Arab negative damping explanation work and lateral instability on
bridges? How did how do you explain instabilities? If the bridge
isn't tuned to the pedestrian natural frequency, is it very special for this London Millennium
Bridge, as some people claim that it just happened to that they picked
a vibration or they picked a suspension mechanism that meant that the mode of vibration
of the bridge was very close to a natural pedestrian frequency. Unfortunately,
that's not the case. You also you've got.
And one of the reasons we know that's not the case is if you dig into the literature, there'd be many examples.
Of bridges, not just the Millennium Bridge that have gone unstable
due to crowds walking on them, not due to traffic, but due to crowds. The Auckland Harbour
Bridge doing a particularly marrie protest, walk a footbridge in Changi Airport in Singapore
and then in Bristol Suspension Bridge, it was observed. So if you know anything about Bristol,
the suspension bridge goes across a gorge which connects the city to Ashton Gate Park.
And every summer there's a Europe's largest balloon festival in the park. And in two of those evenings,
they have a nightclub event. And what they do is they get a massive arena and about 100 balloons get
lit up and lots of novelty balloons get lit up as massive
festivities cetera. And it finishes at 10 o'clock and everyone walks home. Now, the quickest way to
walk home is across the suspension bridge. And they would close it to traffic as well so that you only get people walking
across it. And then it was noted that the bridge started to move. So my colleague John McDonald
was called in and the following year he instrumented the bridge. And this is what he found.
What he found. What I'm showing you here is this is over the course of the week. And
these are the peak lateral vibrations just after the nightclub event.
So these are what the. They are a mess. So the
average displacement, daytime evening, daytime evening and then the nightly event,
massive in the lateral direction, not in the vertical or the torsion.
And if you look in more detail, you can actually look at the record
during the time the pedestrians were walking across. But what was noticed is that
the as the number of pedestrians changed, there appeared to be
a critical number beyond which added, if you can see this in the blue and the red here,
that two different modes of the suspension bridge were excited. So
it wasn't just it wasn't just that the pedestrians happen to be walking at
a certain frequency, which was the mode of the bridge. The pedestrian average frequency
was actually halfway between two different modes, one of which, if you like, is is like
this. And the other of which was like that. And both were excited.
How can that be synchrony? How can you get simultaneous synchronisation to
two different vibration frequencies? There's something odd about that.
So what about Arabs? Negative damping idea? Well, here's a really
simple nonconstructive explanation of their idea, and
I think they hit the nail on the head. Have you ever considered what you
do when you're walking on ground that is moving laterally. And if
you've never experienced this, you probably lucky there are some people in the audience. There are
some experiments that can be done by adults that involve how can I put
this a certain change in their perception due to circlet liquid
intake that makes them perceive that the ground is moving. And if you've ever witnessed, don't try
this at home, children, please. If you've ever witnessed somebody undertaking one of those experiments, they tend
to walk in a way that is a bit sort of like this, is that they're changing their balance all the time
as the ground seems to move away from them and doesn't seem to be very flat.
But seriously, what are you doing? What are you doing? If you are moving, if you're on ground,
that's moving perhaps some I don't know the theme park or something. Sometimes you step onto those rotating
platforms. You walk on ground that's moving. What you desperately try and do is not fall over.
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
footstep loses that momentum. I'm going to try and write myself
to to decrease the amplitude of my oscillation. So whatever is causing
me to oscillate, I don't care about the ground. I don't care about necessarily where I'm headed,
although in a rough sense I might. What I care about is not falling over. And therefore I want to lose
energy from this degree of freedom. I want to decrease their energy. Where do
I want to put that energy? Well, there's two places I can put it. I can warm my muscles up
or I can give it to the ground. I can push against the ground. And that's generally what we do when
we walk. That's where we get extra momentum from when we run, etc. We do work
against the ground. So if I want to not fall over in that direction,
I'm going to push against the ground. And the only place that energy is going to go
is into the ground itself. Now, normally, the ground is very stiff and it can take it,
you know, might get impacted or whatever. But if you're on something like a suspension bridge on average,
what you're doing is it starts to move is you're going to give energy to the bridge.
That's irrespective of whatever speed you are walking at. And whatever
speed the ground is moving, on average, whenever you start to wobble, you're going to
give energy to the bridge. And that's essentially what negative dumping is. It's
it's that the coefficient. It's the degree to which, on average, a pedestrian
wants to give energy to the bridge. If you caused the pedestrian to move and
therefore, if you get enough pedestrians, then they get enough of this negative
damping to overcome the natural bridge dynamics, the natural damping of the bridge. And
the thing will go unstable. And in fact, this is not a mysterious thing. This is completely analogous
to a flutter instability in aerospace engineering and aerospace engineering. If you have
the wind blow fast enough over a wing or indeed the Tacoma Narrows Bridge did something
similar where there was enough vortex shedding that caused the natural dynamics
to be negatively damped. And that gives rise to a flutter instability. This is also known as a hop bifurcation
in mathematics, and it doesn't require any matching of frequencies or any synchronisation.
But how to estimate this negative damping coefficient and this critical number of pedestrians?
So I could show you another video, but I probably won't because I couldn't edit it very well.
But what you can do is you can start to say, well, what actually happens when you walk?
And basically what you need to understand is that there's a difference between the frequency
with which you're putting your feet down and the frequency with which you're wobbling
from side to side. Those are completely independent degrees of freedom.
And you can have a model for telling you when you put your foot down. And sometimes the central pattern generator
has decided I'm going to walk at this pace unless the ground stops me from doing so. But
if I'm walking at this pace and the ground starts to move, then I might not change my pace. I might do
a little bit, but I might adjust where I put my feet. And that's a different motion. The side
to side motion is a different motion from the pattern generator that's
telling me when to put my foot down. They're going to be coupled. And so these. So for simplicity,
we could imagine that I am trying to walk at a certain speed. That's in the so-called
sagittal plane, the four aft plane. And I'm moving
from side to side as this lateral oscillating structure is causing me to do
so in the so-called frontal plane. And that you can describe by various
models of pendulum falling onto another pendulum, a
pendulum. As I fall onto the other foot. And you can get various time
series. So they're upside down pendulums rather than pendulums the right way up.
And you can write down equations for these and I won't go into it. But there's a number of different equations. John
MacDonald's original model just assumes that this is prescribed.
So I'm walking at a certain pace and nothing will change that. Then you can look at a model that allows that
to be adapted such that if I have to take a wider step, then I'm going to adjust
how far forwards. And when I put the foot down. So it's it's just a geometric
adjustment corresponding to how wide my leg is.
And the adjustment is due to a sort of control law that says I'm going to try not to fall over.
And then there's another model that Igor Belak and his student came up with,
which just says, I will put my foot down whenever I feel myself going
over the 45 degrees. Then I'll switch onto the other foot, 90 degrees, 180 degrees. As soon as I
feel myself passing through this. I'll put my foot down. So we
tried all three models. There are more complicated models. There are models of articulated legs. There
are models that fully couple everything. There are models that have multiple degrees of freedom that add masses for the legs as
well as mass for the core. There are models that contain the central pattern generator
as part of them. But in essence, all models of walking have this notion
of I put my weight on one foot, then I lift it and I put my weight on the other foot.
And what we found. So we then did some mathematical analysis and the mathematical analysis
is model independent. It says, As long as my model is like that, I've got a bridge.
Linear oscillator with a massive mass. Just goes back and forwards depending on how it's forced. And I've got
pedestrian's, much less mass, that provide feedback to the bridge. And
the bridge provides feedback to them. And then what we found was that you can
calculate from any model the degree of negative damping, and
it contains three components. I've called them Sigma one, sigma two and Sigma three. So in
the simple John MacDonell model, it's actually the Bosnian town model. Sigma one is the only
thing that occurs, which is the dependence of the average
lateral force on the bridge is velocity. So if the bridge moves a bit faster than I
put my foot out wider. This diagram sort of explains where
the asymmetry comes from, because you might think you might think that on average, this notion
of me putting my foot out less wide or wider to stop myself falling over would balance.
In other words, the number of times I push the bridge in the direction it's already going would
be the same as the number of times I push the bridge the other way. But it isn't and it isn't
for the following reason. If I push the bridge in the direction that it's already going,
I put my leg down and as my leg moves. So this angle increases
and I give a higher component of force in that direction. If I put my leg down here
and the bridge is moving towards me. So that causes my foot to go that way and I put
less force. My force is now going downwards. There is there is now no
component of force in that direction. And so you see there is a fundamental difference between doing
that well and doing that. So you see when people are walking
on, they sometimes do this business that provides much less energy than if you do
that and end up putting a foot out wide and pushing the bridge in the same direction. And that's where the fundamental
asymmetry in this negative damping cups comes from. And then there's a an adjustment
in where you put your foot. And then there's an adjustment in where you put
your foot forwards and backwards as well. And then we run all three models and we found
the same sort of thing. So what am I showing here? I've almost finished. So
this is the I'm going to show you what happens as we do a simulation. Then
add another pedestrian. Then do a simulation. Then add another pedestrian. Then do a simulation.
So up to about 200 pedestrians in this particular model.
We take the pedestrians to all have a similar frequency with a randomly assigned
frequency and face with it, just a five percent variance.
And the pedestrians, this is their amplitude here, doesn't change much
depending on whether the bridge is stable or unstable. But what you see is the bridge
motion here. And as soon as you pass 200 pedestrians, then the bridge motion takes
off. And in this model, there is nothing to stop it. Now, two things you notice.
One is the negative damping coefficient, the damping curve fish out of
all the pedestrians minus that of the bridge.
It changes sign at this 200 and the amplitude starts to grow. And
this here are is the Kuramoto order parameter. Remember, that was that was the degree
of how much the oscillators are in sync with each other. And in this case, there's no difference
between after the bridge amplitude. That's this here grows
very large. And was the bridge amplitude is small. Then we added this feedback
to the model. And the feedback is I change how often I put my foot or
I change how far forwards I put my foot, depending on how wide I am putting my feet.
If you include that feedback, then something really interesting happens. You do see a large degree
of coherence at least, or a change in the order parameter apparent synchronisation.
This is, if you like, 20 percent of the pedestrians appear to be walking in step.
But notice, this happens as a result of the instability. The damping coefficient
goes negative. The amplitude increases. And then as the amplitude increases,
that causes more of this. What's sometimes called phase pulling or coherence
amongst the pedestrians. You can also plot the
here I am plotting the bridge frequency dividing by the pedestrian frequency.
And here I am plotting the damping coefficient. And what you find
is that this damping coefficient can actually sometimes. This is the amount of damping per
pedestrian or the amount of negative damping per pedestrian that actually its frequency
dependent. It's a question of timing. What I was saying about putting your foot out wide and
putting your foot out was kind of assuming that I am walking at a similar pace to the bridge's oscillating.
If the bridge is oscillating much faster or much slower than actually during the time
that I'm on my right foot, the bridge might have moved back. And so there's an element of timing.
So it's actually it is frequency dependent. But what we find here is that between a frequency
ratio of point eight and two of the bridge frequency to the pedestrian frequency, the bridge is liable
to this instability. Because these curves are positive, these three
different curves are for different pedestrian frequencies. So I'm plotting this just as a ratio.
It isn't actually only a function of this ratio. It depends on the underlying pedestrian or bridge
frequency. The blue curve is an analytic curve, which you can derive from the simple model.
And this here is in simulations, the number of pedestrians it took
to make the bridge go unstable. So if there's this large frequency mismatch, you can never
make it go unstable. These are violin plots that show you a big scatter because there's a scatter
depending on all kinds of variations that are here. But essentially, you do not have
to be in this regime where there's a one to one ratio
between the bridge and the pedestrian frequency. And in fact, that's not optimal. About one point three or one point four
ratio were optimal in the sense of trying to trigger the instability. The worst case seems
to be around. One point four is the ratio between the bridge frequency and the pedestrian frequency. Then
it only takes about 150 pedestrians for these parameters.
And we could do the same for the falling over model, the Belic et al model
that I mentioned. And this one also shows a very large amount of coherence.
This one does synchronise and already been used as a as an example of synchronisation. But once again,
synchronisation happens after the instability, the instability, whether negative damping first kicks
in, occurs at 156 pedestrians. And then at least for these simulations where
we're adding the pedestrian slowly. It took a little bit later before the amplitude saturated and the amplitude
saturates as the pedestrians now all start to fall into sync. You see,
prior to the instability, they all have random phases in plotting a sample,
at least of the pedestrians as sine waves. But here they're most of their sine waves or lie
on top of each other. And we can do a similar thing and find sorry,
something's got chopped off here. This damping coefficient in the number of pedestrians. So
to sleights almost there direct conclusions. Partial synchronisation
sync this 20 percent of pedestrians might synchronise, probably is
a property of pedestrians on laterally excited bridges, on laterally moving bridges. So probably
is a thing. It's not a bad theory, but I claim it doesn't
explain the instability. Doesn't have explanatory power. It's an observation
of what happens. Doesn't explain why it happens. And in particular, pedestrians are not like Horgan's
clocks. And the bridge is not a passive coupling device. It has its own
natural frequency. It's not like the pedestrians are choosing to synchronise on their
average frequency. And it requires, therefore, the pedestrian walking frequencies to be
tuned to the frequency of the bridge. And it also gives no insight
to the engineers on how to measure the critical number of pedestrians. In order to
have a safe bridge. This concept to negative damping, I claim,
is intuitive. It's basically what I do in order to not fall over when
I'm walking on moving ground. What I'm doing is I'm giving energy
to the bridge to stop myself having too much energy in the lateral direction.
This effect can be measured. You can measure Arop measured it. You can model it. And
you can use it as part of a simulation. It's a robust it's robust to the choice of mathematical
model. We took three different mathematical models and they all showed this negative damping
and the negative damping showed when the numerical simulations go unstable.
It explains instability across a wide range of Brigg fruit bridge frequencies. It's not just tuned
to the idea that the pedestrians have to be walking at the bridge frequency or indeed that they have to be walking at
similar frequencies to each other. And it gives the engineer a way to
compute this critical number. Protests. Pedestrians. And there's a nor
there, we should say. And the necessary bridge damping to avoid instability.
So in a sense, in a sense, the engineers were right all along.
All this hoo ha! All this stuff about saying, wow, what a great example of synchronisation.
It probably isn't. There was probably some synchrony, but actually a large
amplitude. There always is some synchrony. But it was probably more like these cats that
I have here in that there was precious little synchrony going on.
But that's not the point. The point is, it's this negative damping
criterion, which is something you can calculate is nowhere near as famous. It has 30 citations,
whereas the nature paper on synchronisation has several hundred, maybe a thousand. I'm not sure
it's been there in the engineering literature for a long time. It's just an inconvenient truth. It's
it's if you like, if I'm can be a bit more philosophical. The synchronisation of near identical oscillators
is a great mathematical theory that explains all kinds of things, but it doesn't explain everything.
And in my view, it doesn't explain why and it doesn't predict when
bridges go unstable to lateral oscillations. So maybe maybe this
is a case of what Thomas Huxley described as the great tragedy of science, the slaying
of a beautiful hypothesis by an ugly fact. But maybe
not. Maybe actually, it's a failure to apply
the core principle of mathematical modelling. Which is Ockham's Razor. If
there exist two explanations for an occurrence, the one requires the smallest number of assumptions
is usually correct. Sometimes we go out there with
our hammers as mathematicians looking for nails and saying, oh, I am an expert in synchrony, oh, look, there's a perfect
example of synchrony without actually checking the assumptions, without saying what is
truly oscillating here, what a truly the natural frequencies, et cetera. Or, if
you like, keep it simple, stupid. If I've learnt anything, is a mathematical
modeller. I almost never. Make
progress or have insights by making something more complicated. It's nearly
always the case that the truth is much simpler than you think. You just haven't been
looking at it the right way. And avoid the wisdom
of the crowd, popular explanations are often wrong. There's a there's a physics textbook
explanation. There's another bridge instability, very famous in physics and engineering textbooks and calculus
textbooks that says this is an example of resonance where the wind started resonating
or the vortices shed from the cables of a bridge resonated with the bridge's natural frequency.
Balderdash, or some of the popular explanations you see of how aeroplanes generate
lift or even explanations I've seen for why there is a Nobel prise for economics.
There isn't. And why there was never one for maths, which has nothing to do with
Nobel and Mittagong Lefler and their personal relationship or otherwise.
So finally then, if I can give you some messages, it's think.
Never be afraid to think for yourself. Be a little bit scientifically stubborn.
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
out for yourself. Question. Cheque the assumptions. Apply Ockham's Razor.
What actually are the hidden assumptions. Am I doing here? Am I making this problem more complicated
than I need to? And sometimes sometimes we do that. And I think mathematics
is a very good tool for cutting out assumptions and making things simpler if we only
did it the right way. And so finally, then, I'd like you to remember that rhythmic clapping without
being presumptuous. And thank you very much.