nm0724: that you know [0.2] what's going down [0.2] this morning [0.5] er we 
got some guy where are you from anyway [0.5] 
om0725: Centre for Applied Language Studies 
nm0724: there you are so he's local [0.5] we er he so he's not that [0.2] er oh 
you weren't here we had the Polish T-V lecture in er [0.2] er [0.7] lecture or 
videoing a lecture the other day [0.4] but er [0.3] that wasn't you lot have 
you lot been er some of you have been videoed haven't you [1.0] the oh yeah 
when er some of you were in the lab i know others of you missed it but for er 
[0.4] the N-B-C the American one [0.5] er that went out on the er [0.6] er N-B-
C Today programme [0.7] s-, [0.3] oh no not B-B-C N-B-C [0.2] so it was er [0.
3] i think something like thirteen-million Americans would have watched you er 
[0.8] and there's been feedback saying how good you looked and things like that 
[0.8] er [0.2] and and at least one of you appeared on Sky T-V er or er filmed 
for Sky yesterday but [0.2] oh no names mentioned [2.5] the celebrities in the 
audience it's amazing [0.5] er i got a favour to ask in fact as well [0.3] has 
anybody got [0.4] i know all of you'll be skiving off this afternoon and 
putting your feet up and things like that as typical students do on a Friday 
afternoon [0.4] has anybody got a video [0.4] er machine or equipment or 
anything or have access to it [1.8] you do [1.1] well it's n-, s-, n-, n-, n-, 
noth-, nothing naughty i'm asking [laughter] for no [0.5] er [0.6] i i rushed 
out the house this morning and i forgot [0.3] the [0.2] Newsround tonight [0.4] 
would it be possible [0.2] for you [0.5] 
sm0726: yeah sure [0.5] 
nm0724: brill [0.3] excellent [0.4] 'cause 
sm0726: [0.8] 
nm0724: er i think it starts four- [0.2] fifty-five but i might [0.4] but it's 
Beeb one [0.3] four [0.2] something like that [0.8] if it's poss just to video 
it [0.2] i'll pay you handsomely [0.2] you know er [0.3] while this is being 
recorded [0.4] i don't know what shall we say sort of ninety per cent in the 
exam something like that or [0.4] [laughter] that all right [0.3] okay i mean 
er al-, all right ninety-five okay [0.2] that's the final offer [2.0] just so 
this is this is me being er [0.3] very generous [0.2] so and i-, and if ever 
you say you said ninety-five and you don't get it [0.3] then it's all been 
recorded [0.3] so [2.7] right what it what it is on Newsround [0.2] is oh don't 
panic don't panic 
you've had all this before [0.4] er what it is on Newsround is something to do 
with the Sony dog [0.5] and they came here and [0.5] we of course we're the 
most knowledgeable people on earth about the Sony dog [0.4] Sony excepted [0.2] 
and er [0.3] they wanted a bit of waffle on it [0.4] i think that's what it's 
about anyway it could be [0.5] something completely different [0.8] er [1.0] 
let me let me remind you oh it's all right [0.3] step this way step this way [0.
2] you need a crash helmet for this lecture [0.8] it's er i know it's a bit of 
a [0.6] a bit of a crash course [0.3] the the s-, [0.2] stuff we were doing [0.
2] if you remember [0.4] we have a system that we're dealing with [0.3] and 
we're shoving a step input into it [0.4] step input is switching it on [0.2] 
bing step input [0.4] the pra-, this is practical er [laughter] experimentation 
[0.6] and er [0.3] we're seeing how the system responds and we looked at the 
first type of system [0.3] where it's overdamped so [0.3] the system responds 
the output [0.4] morning [0.3] whatever this is it it it eventually gets up to 
its final value but it doesn't overshoot it doesn't do anything 
funny really it just gets there eventually and that's it [1.4] we then had a 
look [0.4] thi-, this is hopefully how i [0.2] how i left you last time we were 
on [0.4] er was looking at the critically damped case [0.6] and how far did we 
go fine [0.2] so this this you've had a look at this bit so far so if i go over 
this you won't [0.2] or shouldn't be saying what the hell is that [0.3] it 
should be yeah i remember that way back [0.2] beyond the [0.4] days gone by [0.
4] the [0.2] in the [0.3] critically damped case just to remind you the 
critically damped case [0.4] is really this sort of [0.3] mathematical ideal on 
the cusp [0.4] probably never really happens in practice type [0.2] it it's a 
it's an ideal type of thing to get to [0.5] it's sort of er [0.3] in a 
political sense it's the pure communist state you know it's an ol-, i-, ideal 
but it's never er going to actually occur in practice [0.6] er [0.6] here we 
got a system [0.2] output [0.3] the the system is G input is U which is a step 
input [0.4] and where we'd got to [0.4] was [0.2] simply at the point of 
looking at what the output response is [0.8] to a step input [1.0] and er [0.5] 
er again 
what we'd done before well what what i'd done and some of you had listened in 
[0.5] was to look at the sort of extreme cases [0.4] when thi-, this is what it 
looks like the output looks like in terms of time physically what the output 
looks like [0.5] and there are three [0.2] values to it or three parts to it [0.
6] one is the just the gain K is just a number [0.4] alpha [0.2] apart from 
being a Greek symbol just to make it look more difficult [0.3] is also just a 
number [0.5] and and this would just be as it is squared so this is just going 
to be a positive value here that's all a gain value [0.5] we've then got one [0.
3] now this this effectively will be the steady state value when things have 
settled down [0.3] and all we've got are two exponential terms [0.7] which we 
can look at at the extreme cases [0.2] and the extreme cases being [0.2] when 
we switched the thing on [0.4] and [0.2] it's like er time is zero [0.2] it's 
the big bang time [0.3] well in a physical sense [0.2] that is not in a [0.3] 
er well any other sense [0.4] and i've got to watch what i say here haven't i 
'cause [0.2] i mean th-, this is 
this is with language studies you get sort of [0.2] er mystical foreign 
students er [0.5] listening in on this or [0.3] yeah [0.6] 
om0725: possibly [0.2] 
nm0724: okay [0.3] for the so for the [0.3] the Latin American students that 
might listen to this transcript [0.3] the big bang i'm talking purely in a [0.
2] sort of Stephen Hawking sense not in er [0.4] er other [0.5] other 
connotations [0.5] and er [0.4] anyway so actually when when t-, [laugh] oh God 
let's go for it [0.2] when T is zero [0.4] then exponential of zero is [0.9] 
sm0727: one [0.3] 
nm0724: one great [0.4] and when [0.5] this lot is zero because we've got a T 
in there [0.2] that going to zero the whole lot [0.5] of that goes to zero [0.
3] so all we've got when we switch the thing on [0.3] this is when we apply the 
step input at that time we go zing [0.3] then the output [0.4] will be this 
value [0.2] one-minus-one [0.4] and one-minus-one is [0.9] 
sm0728: zero [0.8] 
nm0724: one student knows anyway that er th-, [0.2] yeah one-minus-one is zero 
[0.2] which means that the output [0.2] is zero [0.4] which is pretty obvious 
we switch the thing on and the output's er well i was going to say [0.2] i was 
going to swear but i won't [0.4] er the output is zero [0.5] er but why not [0.
4] but er now let's let's take the other case [0.2] 
when time goes off to a big number [0.3] such as infinity or the end of the 
lecture whichever is first [0.5] and [0.2] in this case the exponential minus 
infinity [0.4] is [2.0] 
sm0729: must be zero [0.3] 
nm0724: must be zero [0.2] it sounds good doesn't it it sounds good [0.3] so 
that must be zero [0.4] and here we've got [0.4] er [0.2] time is off to 
infinity or a big number [0.9] and this bit is down to to zero [0.3] so what's 
[0.3] er [0.3] anything multiplied by infinity is [0.5] 
sm0730: infinity 
nm0724: infinity [0.3] except for one specific case because anything multiplied 
by zero is [0.3] 
sm0731: zero [0.2] 
nm0724: zero [0.4] and er we're on one of these sort of [0.5] parad-, what is 
it a p-, a paradigm or something i don't know what [0.2] or is that some exotic 
drink [0.7] 
sm0732: [0.3] 
nm0724: no all right anyway [0.2] it's zero multiplied by any old thing and T 
is a big number [0.3] so the whole lot's going to go to zero there [0.3] so 
it's because [0.2] that's [0.6] zip all [0.4] so we've got one-minus-zero [0.2] 
which is [0.8] 
sm0733: one [0.3] 
nm0724: one thanks very much [0.5] well we ge-, we're getting there even if 
it's the odd sort of person there [0.3] so in a steady state [0.2] this would 
be [0.3] K-over-alpha-squared whatever the number is [0.2] multiplied by one [0.
3] which turns out to be [0.2] 
K-over-alpha-squared [0.8] so that's the critically damped case [0.4] [0.3] 
that's the steady state value [1.5] and the two extremes [0.4] er [0.4] are the 
ones we've just looked at [0.3] the steady state when time goes off to infinity 
[0.6] or a or a big number [0.3] whichever you want [0.8] er is this number K-
over-alpha-squared which is the [0.3] the inherent [0.3] gain [0.2] of the 
system [2.4] again they're just numbers [0.3] K and alpha [0.3] are just 
numbers [1.0] i mean Kay is also a girl's name i know that [0.2] but for the 
purposes of this thing [0.6] just treat her as a number [0.3] if anybody knows 
[0.7] anybody called Kay [2.3] tonight [0.4] say to her Kay you're just a 
number [4.2] if if she gets upset about that [0.7] you know especially if you 
say you're number fifty-eight or something like that [0.2] she gets upset then 
say no the prof says you're just a number [1.3] say how does he know [0.7] er 
[1.5] and then in the initial [1.2] does anybody know anybody called Kay just 
to get you into trouble here [0.6] just as well [0.2] just as well [0.5] 
sm0734: bloke 
nm0724: don't [0.4] pardon 
[0.6] 
sm0734: there's a bloke called Kay [0.5] 
nm0724: is there [1.1] er [0.2] okay [0.4] well i'll [laughter] 
sm0735: 
nm0724: right yeah well [laughter] the er [0.3] 
sm0736: these Japanese [0.5] 
nm0724: really [2.6] okay let's move on [1.9] [laughter] er the in fact [0.2] 
not that i'm going to go through 'cause [0.2] you know we got language students 
who don't know what all this derivative stuff is [0.3] but in fact the 
derivative [0.2] er of the output [0.3] when T is zero [0.4] also [0.4] turns 
out to be zero [0.2] you know we had that we had that little trinket [0.2] 
cropping up before and here it comes again [0.2] it's not particularly 
important no big deal [0.5] all it means is when [0.4] you er [0.3] er er it's 
it's very much like [0.5] er [0.4] er a member of the opposite sex [0.2] when 
you actually suggest something to them [0.2] it takes them a bit of time to get 
going [0.2] you can't suggest something to them and they're off straight away 
[0.2] and the same with this system [0.2] so in fact the [0.2] the other person 
could be a critically damped system [0.7] er [0.8] and in this case so the 
derivative is zero the takes a bit of time to burst into life essentially 
that's all it means [0.5] what this critically damped [0.4] system looks like 
[0.4] there's no overshoot [0.3] 
the dotted this this dotted thing here [0.3] is the steady state value [0.3] er 
this is the output [0.5] here's time along here [0.3] we switch the switch [0.
3] and up it goes [0.3] but it doesn't actually [1.3] overshoot [0.4] in theory 
[0.2] it doesn't overshoot the final [0.2] value that it settles down to [2.3] 
in fact it is the most rapid [0.5] response the critically damped thing what it 
means is it's the most rapid response that you can possibly get on the face of 
this earth [0.6] without [0.6] any overshoot [3.8] for the overdamped case you 
remember [0.3] it [0.2] it [0.2] didn't overshoot it er [0.2] came in from once 
i eventually got there [2.6] and i've said here it's desirable this particular 
[0.3] goal this particular type critically damped response [0.4] if you're 
looking at machine tools [0.5] er or robots [0.2] in fact or er a lot of 
production machinery [0.2] welding machines is a a prime example [0.7] i know 
tonight you'll all go home and play with a welding machine [0.5] as you do [0.
3] and with that welding machine the aim the target [0.4] is to make the thing 
critically damped [0.5] so [0.6] if if you can imagine for a moment [0.2] that 
i am a welding machine [0.4] and [0.3] this the top of the projector is the 
thing [0.2] like what i am supposed 
to be welding [0.4] then [0.2] here's my little torch [0.2] or stick or 
whatever the welding thing is [0.4] and i the design of it [0.2] i want this 
arm to come in as quickly as possible [0.7] do do you need a v-, a vocal 
commentary [0.4] he moved his arm [laughter] in the direction of [laughter] so 
i want the arm to come in as quick as possible [0.2] to to that point zing like 
that do the do the welding and move away again [0.3] now if it's overdamped the 
case we saw before [0.2] it will not come in zing it will come in zing and 
eventually get there [0.6] if it's as the case we're going to move on to [0.2] 
it goes past it overshoots [0.3] then of course it would go [0.3] zing and we 
messed the [0.2] and you can imagine whatever it is we've gone too far [0.5] 
and know what going too far gets you [1.0] er so this critically damped case is 
the [0.3] the [0.2] the perfect the ideal case if we had [0.3] a perfect welder 
[0.4] see knows all about perfect welders he can afford to miss the first ten 
minutes [0.4] if we had the perfect welder [0.3] it would be critically damped 
[0.3] however [0.5] i this this thing here [0.5] is important [0.6] that 
that's really in theory [0.2] this is the theoretical [0.2] for Christ's sake 
if you get a job [0.3] at the end of this course [0.3] as a [0.2] welding 
machine setter upper [0.3] or whatever it happens to be [0.2] or someone who's 
in charge of people who set welding machines up or [0.2] or whatever [0.4] er 
then [0.8] don't [0.3] do it [0.2] quite this way [0.3] aim for this as a 
perfect thing but remember [0.4] probably you're going to get a bit of noise 
floating around probably you're going to get a bit of variability [0.4] one [0.
2] robot's welding stick [0.2] might be a little bit longer than another 
robot's welding stick [0.3] you know in this world [0.2] robots have welding 
sticks of different lengths we have to be er [0.4] aware of that [0.4] and that 
might mean if you've set the thing up [0.5] to come in just from one side the 
the critically damped case [0.5] it [0.2] it might [0.3] bang the [0.3] so you 
might break it it might [0.2] you you could go too far [0.4] er the other thing 
with noise [0.2] really you may not want this to overshoot and er with a bit of 
noise on the top [0.3] the si-, the signal wouldn't look like that it would 
come in [0.3] 
and it it might go like which means you would in fact be [0.3] overshooting a 
little bit [1.0] so in practice what happens is [0.9] er you would slug it down 
[0.3] a bit [0.3] so in [0.6] a a lot of manufacturing cases in fact it's 
really slugged down a hell of a lot [0.5] er [0.4] typical production line down 
at [0.3] say Ford [0.2] who are [0.3] about to [0.7] close the production line 
down i don't know whether that's any coincidence [0.5] but so at Dagenham the 
robots there they have some big KUKA robots that do welding and parts 
manipulation and so on [0.4] and they're slowed down to about a third [0.3] 
sort of three times if you like [0.2] slower than they could actually [0.4] 
operate at [0.3] for safety reasons for that [0.2] reason that if they operated 
faster than that [0.5] then [0.3] they could the the to achieve [0.3] more 
production because if the things are operating faster they get more things 
going through more welds three welds a m-, a minute instead of one weld or 
whatever it happens to be [0.5] but as you get towards that optimum this 
critically damped case which is what we're looking at [0.5] you've got 
all sorts of potential problems because of the variability in what you're 
dealing with [0.5] so this is a an ideal a theoretical case to aim for [0.3] 
here's the theory [0.3] now let's detune slow things down let's come back to 
the overdamped case a bit in a practical way [0.4] in order that things don't 
blow up and we don't break things and so on [0.2] so forth [0.3] so that's what 
we're about [1.4] so there er critically damped is a sort of [0.3] theoretical 
ideal [0.8] er it's a bit like my wage packet it's completely theoretical but 
the [0.9] it's not ideal either [1.3] it seems seem that the language [0.2] 
they get paid loads of dosh over there so [0.8] they get p-, [0.3] get paid it 
all in foreign money and er [0.2] [laughter] [0.6] right [0.6] er we've done [0.
6] the overdamped case [0.5] we've done the critical damped [0.2] case [1.1] 
and er [0.4] we're going to have a look at the [0.3] this is this is in fact 
this is a really sad moment [0.4] because this underdamped response [0.5] this 
is it this is the final thing we're going to be covering on the course [0.4] so 
[0.6] i know i know it's a time for holding back the tears for [0.7] er you 
know [0.8] 
stifling [0.4] not a yawn [0.2] but [0.8] stifling something or other [0.9] er 
so the o-, the the overdamped [0.2] it didn't didn't overshoot critically 
damped's right on the cusp [0.4] underdamped is the the final thing we're going 
to look at [0.2] and it's called underdamped because it's underdamped [0.5] er 
[0.3] the the damping [0.3] is not enough [0.4] to stop [0.3] overshoots [0.3] 
essentially [1.0] and this this type of response the output [0.4] overshoots 
its steady state value [0.4] so i'll just [0.2] flick back [0.4] to this [0.2] 
previous overhead [1.1] he put the previous overhead on the projector [0.5] 
flick back to that [0.3] then in this case we was the the sort of ideal in the 
n-, furthest we could go [0.2] quickest we could go without actually 
overshooting here [0.2] we're actually going to go past this final value [0.3] 
or [0.3] er for language students from [0.3] er the southern English [0.3] past 
the final value [1.1] er [0.2] the transfer function roots in this case you 
remember we had [0.5] the first case the overdamped case the roots of the 
denominator [0.5] were [0.3] were like two [0.2] different [0.2] roots as it 
were but real [0.2] roots [0.7] and in the [0.6] 
critically damped case [0.3] the roots were in fact exactly the same [2.6] and 
in this case [0.3] for the underdamped case [0.3] we've got [0.3] complex 
conjugates [0.2] we've got er [0.6] er [0.2] a real part and an imaginary part 
[5.3] and and in this case for the underdamped response [0.4] the output [0.2] 
oscillates [1.0] following a step input [2.5] oscillates meaning it it goes 
backwards and forwards [3.6] and just [0.3] just to see what i'm getting at 
here what we're talking about here [0.5] an an experiment for tonight is [0.3] 
trying to get the [0.2] key into the door of your room or house wherever you're 
living [0.7] er [0.3] for a [0.3] an overdamped case just just go normally [0.
3] just go and you should be all right to go in fact that's probably critically 
damped isn't it just go straight to the keyhole turn the door and so on [0.8] 
just for the purposes of experimentation [0.2] have a few drinks [0.4] and then 
try [0.2] and get the key in the door [0.2] and you'll find [0.2] more than 
like well i find anyway i don-, [0.3] is is that the key you'll have to sort of 
j-, j-, jiggle it around a bit like that [0.3] and eventually you get 
the key in the door [0.3] the more drinks the have [0.2] you have the more 
jiggling around that you have to do [0.3] until if you have [0.3] too many i'm 
not suggesting you have too many but if you have too many [0.3] probably goes 
completely unstable and you'll fall over sideways while you're trying to 
[laughter] get [0.2] the key in the door [0.6] but this one the oscillations 
are they're trying to to get r-, right on to the then you over you go past and 
you come you know the [0.2] you know the feeling you go [0.2] it's a [0.2] it's 
a bit like my wife's driving so [0.9] [laughter] [cough] [0.7] don't laugh it's 
true unfort-, [laugh] [1.0] that's another another rear reflector i shall be 
mending tomorrow [1.7] er the amount of damping [0.8] dictates [0.3] how 
quickly these oscillations die down [1.1] a lot of damping [0.2] the 
oscillations don't hang around for very long [3.3] not much damping and they 
stick around for quite a while [5.7] and in the limits [0.4] if there was no 
damping [2.2] then [0.3] it's [0.2] purely [0.2] it's just oscillating [0.5] 
off it goes [1.7] just oscillates [3.6] and for the foreign language students 
[0.3] very much like a Citroen Diane suspension [1.0] 
it's getting [0.2] towards that [5.5] or le Diane de Citroen [0.2] or whatever 
it [laughter] [1.6] whatever it is [1.0] am i supposed to interact with this or 
should this be a [0.3] 
om0725: 
nm0724: [laughter] [0.4] just that no i mean normally this is deadly serious is 
these lectures [0.3] are they not [laughter] [0.2] and er [0.3] 
sm0737: compared with your other ones [0.2] 
nm0724: compared with other ones yeah 
sm0738: [0.3] 
nm0724: that's right so this is i mean there have been no Skoda jokes today at 
all [1.8] they can't remember any 
sm0739: opportunities for one [1.1] 
nm0724: oh i don't need opportunities [1.3] 
sm0740: [0.2] 
nm0724: do you want a Skoda joke [0.9] 
sm0741: 
nm0724: yeah [0.6] i don't know have i had any Skoda this is ah this is for the 
[0.2] Czech [0.2] listeners er [1.2] my my wife is Czech this is a true story 
[0.2] actually [0.8] and er [0.9] w-, years ago when she first came to England 
[1.0] er [0.4] the [0.2] university photographer caught us at the degree 
ceremony [0.3] and said aha you're Czech i've got a Skoda joke as people do [0.
2] tend to do [0.7] and is i-, you may have heard this bit but stick with it [0.
4] and he said er [0.4] why why do Skodas have a heated rear windscreen [1.0] 
and my wife er [0.4] and he said so you can keep your hands 
warm when you're pushing it [1.3] [laughter] and [0.3] we we sort of [0.4] well 
me and another guy we were sort of [0.2] laughing a little bit [0.2] 'cause we 
haven't heard it before see it was many years ago [0.7] but my wife didn't 
laugh at all [0.7] i thought well maybe he's offended her wh-, wh-, why aren't 
you laughing it's quite funny [0.4] and she said well Skodas don't have a 
heated rear windscreen [1.3] [laughter] [laughter] [1.1] right [0.3] way to go 
[0.2] [laughter] [1.0] the true jokes are the funniest [0.7] er as the damping 
is in-, sorry that's got nothing to do with the lecture but what the hell [0.5] 
as the damping is increased so we get [0.4] to the critically damped case [0.2] 
the case [0.2] like what we were at [0.3] before [1.3] so no damping it's just 
[0.3] purely oscillatory [0.2] increase the damping increase the damping [0.3] 
and we will get [0.2] the critically damped case where you don't get any 
overshoots [0.3] no oscillations at all [2.9] i just put the what a shame 
'cause that's what i felt like [0.7] that's an optional extra as far as your 
notes are concerned [0.4] if anybody in the exam [0.3] puts something like what 
a shame [0.3] then well [0.2] it's up 
to you isn't it [1.9] the decision is yours [0.3] [2.5] [sneeze] bless you [4.
1] has anybody seen Bicentennial Man [0.7] yet [2.1] no [2.0] okay [1.1] so y-, 
[0.3] note we are with science students engineering students [0.2] we're not 
into arty things like [0.4] Robin Williams [0.2] films and stuff so has every-, 
everybody finished with that [0.4] 
sm0743: [0.6] 
nm0724: okay [1.6] we wait while the slow student catches up [laughter] [0.5] 
well apparently it's about [0.2] some i i only know this 'cause some 
Argentinian journalist contacted me to say what do you think of it [0.4] as 
Argentinian journalists do [0.3] 
sm0744: based on the Asimov story [1.0] 
nm0724: i think it's yeah it this robot 
sm0745: about the robot that wanted to be human 
nm0724: th-, [0.3] that's right it couldn't be human and wants to have human 
feelings and stuff like that [0.5] so the journalist asked me [0.3] can robots 
have [0.6] feelings and so on [1.2] [0.2] that's why i've put what a shame that 
was the link putting what a shame 'cause the system doesn't [0.5] have [0.2] 
oscillations [3.8] [laughter] that's it [0.9] let's go for the next one [0.3] 
now unfortunately [0.2] you're not going to get away with this lightly [0.3] i 
mean normally because we don't do stuff like maths and 
things like that i mean [0.2] but we have to because we've got [0.4] er 
language student listening in [0.2] we have to do some tough stuff [0.2] okay 
[0.5] so er [0.3] purely for this purpose we'll [0.2] throw some equations and 
maths on the board just so it looks [0.2] as though we actually do some work [0.
6] and then they'll go away thinking cor i'm glad i didn't do that degree 
that's blooming hard that one [0.8] wh-, how the hell do they know all that 
stuff [0.8] and then once once we're not recording then we'll get back to the 
easy life that we normally [0.8] do [0.5] [laughter] [0.4] like 
Skoda jokes and things like that [0.5] right [0.2] just to remind you we're 
still we're still looking at the what case are we looking at the underdamped 
case that's the one thanks for that [0.4] and in this [0.5] instance [0.2] we 
have got [0.3] complex conjugate roots [0.2] [0.3] so the [0.3] the transfer 
function G [0.3] that relates input to output [0.9] has this gain K [0.2] there 
she is again on the top line [0.2] or he again [0.2] or he again [0.4] and on 
the [0.3] and on the bottom line [0.3] we have [0.6] S-plus-alpha-minus-J-beta 
[0.9] bring in another Greek symbol [0.2] make it more complicated [0.6] for 
the Greek listeners this is probably all right though [0.5] and and multiply by 
S-plus-alpha- [0.2] plus- [0.2] J-beta [2.5] and you can see [0.2] here why 
they use the term J [0.3] and not [0.3] some word like egg [0.4] because if 
they called J egg [0.3] then we'd have egg beta [0.2] and that would be a bit 
silly [laughter] [0.4] see there's there's logic in these symbols you thought 
why they use J now you know [0.2] now you know [0.8] er [0.4] so we've got 
complex conjugate routes [0.2] i'll just [0.2] just to make it look again for 
the language people listening in [0.2] 
make it multiply things out and so on [0.4] and we can also get [0.3] if we 
multiply this stuff out to get rid of the J [1.5] or egg if you like whichever 
the alternative [0.8] er [0.3] just i thought i'd scramb-, scramble it up oh 
God [laughter] no no no that's enough [0.3] that was almost a joke there [0.8] 
so scramble it up [0.2] and this is just looking at the denominator [0.5] 
forget K's forgotten for the moment [1.3] poor girl [1.0] er and if we can also 
if we multiply it out [0.3] because this this little bit this little [0.5] 
Frankie here [0.2] is going to crop up again he's going to we're going to go 
through a few and er [0.2] and whoo in it comes [0.2] just to surprise us in a 
little bit [0.3] so it's important to know where we're going [0.2] S- [0.2] 
squared plus two-alpha-S [0.3] and then alpha-squared plus beta-squared so [0.
2] multiplying the bit alpha-plus-J-beta alpha-minus-J-beta [0.3] we're just 
getting rid of that er complex conjugate thing [0.3] or lo and behold we can 
even look at it in one yet one other way [0.8] which is [0.2] this is in fact 
also S-plus-alpha- [0.3] all-squared [0.7] plus beta- [0.4] all-squared [0.7] 
[4.0] A-K-A [0.2] this form [2.4] and 
now those astute members of you that are wide awake unlike myself [0.2] will 
note that this S-plus-alpha- [0.2] all-squared [0.3] is exactly what we had [0.
2] for [0.4] the previous case [0.2] the underdamped case we had [0.2] didn't 
we [0.2] didn't we have something like that [0.2] the critically damped [0.2] 
critically damped case [0.3] for the critically damped case [0.2] we had two 
roots [0.3] exactly equal [0.4] we did somebody's somebody's nodding yeah we 
had something like that [0.4] well it might not have been alpha it might have 
been A or something but [0.3] but we did did we have alpha [0.6] we did [0.3] 
the lectures actually tie together loosely [0.5] so [0.2] S-plus-alpha-all-
squared was what we had before [0.2] and now we've got this extra term the beta 
[0.2] that's the only difference really [0.2] is this extra bit which [0.3] 
when we're factorizing it above [0.4] er we can only do it in terms of the 
complex conjugates [0.2] so that's where we are [0.6] apply a unit step input 
[0.3] this u-, here's U [0.3] unit i've [0.2] just to remind you it's just unit 
single value [0.2] if it was [0.3] five units then we'd just multiply the whole 
thing by five [0.5] 
just to see what happens [2.5] and the output then [0.3] is just the multiple 
of G [0.3] and U [0.4] or G operating on U [2.9] so we've got Y here's [0.2] 
here's G [0.2] K over [0.4] S-plus-alpha-all-squared plus beta-squared [0.8] 
and then multiplied by one-over-S the step [0.6] input thing [2.0] business [2.
8] [4.1] and now [0.8] this is the bit [0.2] when you close your eyes and just 
write and hope for the best [0.5] and then we'll get back to the [0.3] the 
easier stuff [0.6] you know what we did before [0.4] with partial fractions and 
expanded out [0.3] and so on and so forth [0.9] you ready for this [0.8] say 
yes namex we're ready for it [1.0] 
sm0746: yes namex we're ready for it 
nm0724: thank you very much [0.4] here we go [0.3] there we go [0.4] i've [0.4] 
i spent [0.3] all morning working this one out [0.2] before i came in [0.5] 
sm0747: 
nm0724: so that's right this is this is the one earlier [0.5] and er you just 
need a [0.2] squeezy bottle and three matchsticks [0.5] and it all works out [0.
7] so the output [0.3] is [0.7] he said describing the difficult equation [0.4] 
the output is K [0.3] over alpha-squared-plus-beta-squared [0.2] now all of 
these K alpha beta [0.3] are just numbers [0.8] a [0.3] typical number [0.4] is 
three [1.3] so [0.3] they might all be three [0.3] in fact [1.8] 
and in fact if K was three she might feel a bit happier than if she was fifty-
eight [2.3] and then inside [0.3] the [0.2] parentheses [0.7] or brackets or 
whatever you like to call them [0.4] we have [0.2] one-upon-S [0.6] which in 
fac-, as er [0.2] you'll note is something to do with the steady state value [0.
2] and it turned up up there for the unit step input [0.2] and then we have two 
terms which look absolutely awful [0.4] the fir-, both of them [0.2] have the 
same [0.3] denominator the same S-plus-alpha-all-squared-plus-beta-squared so 
that [0.3] that's the denominator on both of those [0.5] the first one is S-
plus-alpha [0.6] and the second one is alpha [1.1] why the hell have i done it 
that way [0.4] because i've sort of cheated a little bit and looked ahead a 
step [0.4] er what i'm trying to do with this [0.3] is to put them in [0.3] 
partial fractions [0.2] into a form [0.3] that you can just go to a book and 
look up how they are [0.3] and it tells you what [0.2] that actually means in 
terms of time [0.5] signal representations [0.4] so really you can go and look 
up what sort of forms you can get to [0.3] and then try and 
chop [0.5] chop this thing up [0.4] er into a form that [0.3] actually gives 
you something that you can just look up [0.2] so you don't have to do any fancy 
stuff [0.6] er [0.3] don't worry when it comes to [0.2] examsville [0.4] this 
[0.2] i'm not going to ask you [0.2] shove this into partial fractions and look 
it up in a book and tell me what [0.2] the answer is [0.2] 'cause i'm assuming 
you'll all be able to do that sort of stuff if you really wanted to [0.5] 
eventually [0.4] er [0.2] so that is not going to crop up [0.2] so here it is 
but don't worry don't think blimey how the hell am i going to learn that stuff 
[0.5] er don't [0.2] panic [0.2] or anything [0.2] it's not going to [0.4] 
you're not going to be asked that sort of thing [0.3] you're going to be asked 
lots more difficult questions than that [0.3] but not this particular thing [0.
7] what does this which is more important [0.2] what does this actually look 
like [0.4] in terms of a time signal [4.6] which is the important thing [1.7] 
time is of the essence [1.5] what the hell does that mean [1.2] i thought 
essence was sort of [1.6] some spice or something like that [0.5] 
what's time got to do with being spicy [1.6] 
sm0748: [0.4] 
nm0724: no [0.4] perhaps it's something to do with Geri Halliwell i don't know 
[1.7] right [1.7] or maybe that's just wishful thinking again [1.1] this is 
just to confuse er all of you never mind those listening in [0.3] here's the 
gain so this is the output [0.3] in terms of time [0.9] this is what it what it 
actually looks like what the syste-, for a [0.3] an underdamped system [0.4] 
what it would actually look like the response that you get [0.9] would would 
appear like [0.4] if it is [0.3] the Citroen Diane that i spoke of earlier [0.
3] it would be the position of the shock absorbers after a [0.4] a bump or 
something like that after a hump [1.2] or hump's not a good word to use [1.3] 
[laughter] [0.5] not in a Citroen Diane anyway [1.7] [laughter] i don't know 
might be possible [4.5] this [0.2] probably if this [0.4] tape gets sent to the 
Vice Chancellor this will be the last lecture [0.3] that i'll be giving 
[laughter] [3.9] there'll be [0.4] women's rights campaigners [0.4] men's 
rights campaigners Skoda rights [laughter] campaigners there'll be all sorts 
[laughter] [0.3] er [1.0] the anti-, [0.2] anti-Slovak lobby or something 
probably the [0.2] 
probably some refugees will crop up somewhere [laughter] in there as well [0.7] 
er [0.3] K all complaining [0.3] K over alpha-squared-plus-B and then there'll 
be people complaining that they weren't included [0.3] K over alpha-squared-
plus-beta-squared is the same gain that's just come through [0.2] the whole 
thing just like a dose of prunes there it is again [laughter] nothing's 
affected it whatsoever [0.5] er [0.2] and then inside the bracket here's that 
one which was the one-over-S there it is again [0.2] and two terms [0.4] er [0.
5] exponential- [0.2] minus-alpha-T [0.2] multiplied by the cosine of beta-T [0.
6] and the same sort of thing again exponential-minus-alpha-T multiplied by the 
sine of beta-T [0.7] and all we got is that that's just m-, little gain A over 
alpha-over-beta [0.5] which are just numbers remember alpha beta and K are just 
numbers [0.6] and the nice little thing here you'll see the alpha and the beta 
[0.5] i find these little things nice [0.4] the exponential has the alpha bit 
in it [0.3] so there's no beta in the exponential [0.4] and the beta turns up 
[0.4] with the sines and 
cosines the frequency type of the the changing element [0.3] and there's no 
alphas [0.2] knocking around with the betas [0.4] so the alphas and the betas 
are sort of er separated [0.4] er and they they turned up in [0.2] a different 
form in the [0.4] the denominator [0.3] and here they're having different 
effects in the output [0.3] but [0.4] can we can we see that in another way [0.
2] well we can this this will [0.5] hopefully make it a little bit clearer [0.
4] the only thing [0.4] i mean it may be first instance look a little bit more 
complicated but in fact it simplifies [0.6] when we're looking at what happened 
[0.4] this can be written by [0.4] playing around with so-, sine and cosine [0.
8] what do they call trig identities or something like that is it [1.2] [1.3] 
sounds like something out of Only Fools and Horses whatever [2.5] and we've got 
the gain that's still there it's sort of lingering in there it is [0.4] K over 
alpha-squared-plus-beta-squared no change there [0.4] and then inside the one 
is still the one and all they've done is play around with these two 
terms messing around with the exponentials and the sine [0.5] er [0.2] i've 
cheated a little bit because [0.4] there should be some sort of phase [0.3] 
shift [0.3] that appears [0.5] in the ins-, it's not really just beta [0.3] 
there should be a phase shift [0.4] but i so when you look at in in years to 
come when you're ruminating on it with your cocoa one night [0.4] and you're 
you're looking at this think blimey this should be a phase shift he's got it 
all wrong [0.2] well [0.4] i i'm just excluded it [0.3] or execu-, got rid of 
it just for the purposes of going through this [0.3] there is a bit of a phase 
shift in there [0.3] but it's not really important in terms of [0.3] what how 
we're looking at this [0.6] er later on [0.4] it may become important but not 
at this stage [1.5] so we've got the gain [0.4] er that that one is still there 
which multiplies by the gain and then we've got this time dependent thing [0.4] 
that in here is [0.2] the square root [0.4] of [0.2] alpha-squared-plus-beta-
squared [0.3] on the top [0.4] divided by beta [0.6] that's that which again is 
just a gain component just a number [0.3] 
and then we've got the two terms [0.3] exponential-minus-alpha-T [0.3] 
multiplied by sine-beta-T [0.9] and again the exponential has the alpha [0.4] 
and the sine has the beta [0.3] and the two things are separate [0.4] so [0.4] 
what does this all look like [0.3] er well couldn't be bothered to work it out 
myself [0.3] so i thought [0.2] let you do a bit of work i thought make this an 
interactive lecture [2.4] the steady state value [0.5] er this is as time goes 
off to infinity [0.7] again [0.2] when time is B [3.6] well what is it [0.3] 
can we work it out [0.2] can we see [4.6] 
sm0749: [0.3] 
nm0724: oh you've given it away [3.1] yeah [0.3] if [0.2] if you set [0.7] T to 
be zero [0.4] in that lot [0.7] then [0.3] the overa-, it doesn't really matter 
what the sine is doing that's going up and down as sines do [0.5] but this this 
thing exponential of [0.4] minus-infinity will be zero [0.4] and that wipes the 
floor with anything else [0.4] so er as time goes off to infinity we've got 
this gain [0.3] multiplied by one-minus-zero [0.4] and one-minus-zero is [1.2] 
sm0750: one [0.5] 
nm0724: one who said zero [0.7] one [0.2] yeah [1.0] so it's just K [0.5] over 
alpha-squared-plus-beta-squared is the steady state value [2.2] in fact [3.3] 
in fact if we 
went [0.6] back [0.8] just out of interest [0.2] if we went back here [0.5] and 
had a look [0.4] at at this thing [0.3] if er we remember with the step input 
[0.2] [sneeze] if we just bless you [0.2] if we just set S to zero [0.2] and 
multiply this thing out [0.4] then which is [0.2] alpha-squared-plus-beta-
squared K over alpha-squared-plus-beta-squared [0.4] so we we can just look at 
the thing in the first place [0.3] and if you remember the steady state value 
for a step input [0.3] we just set S to zero [0.3] and we've got our answer and 
that's the same doing the same thing there [0.5] 
and that's what we've got [0.3] K over alpha-squared-plus-beta-squared [1.4] 
but let us this this denominator [0.2] i threw in [0.2] er a few minutes ago [0.
3] just to [0.4] to add a bit of [0.2] spice and [0.2] ele-, some-, something 
there 'cause it's it's something [0.3] we can sort of ask exam questions on 
sometimes [0.5] and i want to look at it a little bit more closely [1.7] so let 
us [0.8] let us return to the denominator [3.5] the denominator is the bit on 
the bottom [2.6] you should know [5.8] the numerator's the bit on the top [1.6] 
and in the middle well [0.2] keep your eyes closed [1.4] you've got to draw the 
line somewhere there [1.3] ha ha ha ha [laughter] [0.7] so the numerator [2.4] 
is commonly thi-, [0.2] this is the bit on the right hand side [0.4] see i writ-
, i wrote the right hand side first just [0.4] so you what the hell's he doing 
[0.5] and on the right hand side this is what we had before [0.3] S-squared 
plus two-alpha-S [0.3] plus [0.2] alpha-squared-plus-beta-squared that's what 
we had before [0.2] for the denominator [0.8] er [0.3] 
well about [0.4] ten lines ago [0.3] was it not [0.3] was it not [2.4] and [0.
5] this is just to say that [0.8] this this left hand side is how it is [0.3] 
often written [1.3] and we'll see why in a second [0.5] so the S-squared [0.4] 
has not changed it's S-squared [0.5] and then we've got plus two- [0.7] zeta- 
[0.8] omega- [0.3] baby-N or subscript-N- [0.4] S [0.6] plus omega-N-squared [2.
5] and again [0.2] zeta [0.2] we're trying to bring in [0.2] people [0.3] half 
the names that are nearly married to American film stars things like that [0.5] 
so it sort of adds a bit of artistic licence to what we're doing [0.6] so two-
zeta-omega-N-S [0.6] plus [0.3] omega-N-squared [0.6] and we can see that i 
mean straight even i [0.3] could do this [0.2] this morning [0.3] with before 
my coffee [0.6] zeta is [0.6] if we just look at this we equate [0.5] S-squared 
equals S-squared [1.2] there's a profound statement [0.5] and then if we equate 
the S terms [0.4] two-zeta-omega-N [0.5] is two-alpha [1.2] and the twos [0.4] 
cancel [2.0] which leaves us zeta is [1.0] alpha-over-omega-N [0.2] if we want 
[0.4] and also the omega-N-squared is alpha-squared-plus-beta-squared [1.3] 
that's just so [0.2] so what have we got this is where we came from 
the right hand side [0.3] and all i'm saying here is well [0.4] you know if you 
go out in the big world the wide boys down Oxford Street [0.2] they wouldn't 
talk in terms of alphas and betas [0.3] they would talk in terms of zetas and 
omega-Ns [1.0] well they maybe wouldn't [0.2] even talk like that [0.2] but 
that that's the the common terminology is that one [0.5] and as we can see [0.
2] let us let us look [0.3] at extremes let us look at extreme cases [0.4] what 
for instance [0.3] happens [0.2] when beta [0.6] tends to zero [1.3] what goes 
down what happens [2.6] and it's er [0.3] it's intriguing really isn't it when 
beta goes to zero [0.3] 'cause as beta goes to zero [0.7] if you're looking at 
at this side here [0.7] we're we're back to the critically damped [0.3] case [0.
4] when beta heads off as beta gets to zero we're back to the critically damped 
case [0.5] er the roots are just [0.5] S-plus-alpha- [0.6] squared [0.5] all-
squared [2.4] just like what [0.3] they were [0.2] sort of thing before [13.0] 
and and now for the [0.2] the foreign listeners sorry [0.2] maintenant [0.4] er 
nous [0.3] will set zeta [0.6] to un [0.5] [laughter] [2.8] and there it is [1.
1] le damping ratio [1.0] [laughter] [0.3] so zeta 
zeta is actually known [0.8] A-K-A [0.3] the damping ratio [0.9] and is as we 
set it to one [1.1] it's it's all the same thing here [0.2] if you look here [1.
4] it's all magic [0.3] beta goes to zero [0.2] so omega [0.3] tends to alpha 
[0.6] and as omega tends to alpha so zeta [0.2] alpha-over-alpha [0.6] is one 
[1.2] here so [0.2] just just from these equations [0.2] beta to zero [0.6] 
omega [0.2] becomes alpha [0.2] effectively [0.6] alpha-over-alpha [0.4] is 
just is one [0.3] so zeta tends to one [0.3] so this thing [0.2] the damped 
zeta is known as a damping ratio [0.3] and as beta goes to zero [0.4] zeta the 
damping ratio goes to [0.3] to a value of one [1.4] and [0.2] the other and and 
really what beta [0.5] was linked [0.2] to the sine term [0.3] the sine term 
going er really the oscillations [0.4] so what this is saying effectively [0.3] 
is as the oscillations die down [0.2] as beta goes to zero the frequency of 
oscillations dies down and so on [0.4] the damping ratio tends to one [0.3] and 
we tend to the critically damped case where we don't have oscillation [0.5] so 
it all sort of ties together really [1.1] the er another extreme as alpha [0.4] 
tends to zero [9.5] then [1.3] we can see see from here as 
alpha's zero then omega-N [0.5] becomes beta [0.3] or omega-N-squared is beta-
squared [2.0] and this omega-N [0.3] which is why we've got this subscript-N [0.
8] is also known as the natural frequency [8.2] and for this [0.2] for this 
reason effectively [0.5] er [2.3] as beta goes to zero [0.4] er as alpha goes 
to zero omega-N [0.2] tends to [0.2] to beta [0.3] becomes [0.2] beta [0.5] and 
[0.3] er with alpha being zero it means zeta [0.5] is zero [0.9] so and zeta 
being zero we've got no damping [0.3] we've we've gone in the direction of the 
[0.3] pure [0.3] oscillations [1.3] we we switch the thing on [0.3] and it just 
oscillates it's a pure oscillator [2.2] it just goes up and down up and down up 
and down [0.2] never gets exhausted never gets tired [2.0] never dies down [0.
2] just keeps going [3.3] it's a bit like Bolton Wanderers [0.2] really [0.2] 
just keep going up and down [0.4] never [1.0] never settle down [1.1] and er 
sm0752: [0.6] 
nm0724: pardon [1.8] oh [0.2] [laugh] [0.3] you're not a Bolton fan are you [1.
3] sorry about that i didn't [1.0] as opposed to namex which is just going down 
and down so [3.0] and so for the underdamped case this is the whole skaboodle 
[0.7] this term zeta [0.8] will [0.4] go from zero [1.0] in the case 
when there's no damping we've just got pure oscillations [0.4] to one [0.3] in 
the case where we were at before [0.2] the critically damped case [0.7] and [0.
2] realistically as the term zeta [0.4] tends towards zero so the oscillations 
get bigger there's more of them [0.4] as zeta gets w-, towards one [0.5] then 
the oscillations get smaller [0.3] so so [0.2] zeta is just a [0.3] a knob that 
you can twiddle [0.2] effectively [0.3] and between zero and one [0.3] and as 
you twiddle it [0.4] so oscillations get bigger or smaller [1.3] just just to 
conclude this and then we'll have a [0.5] er a breather [2.3] it's really 
bringing things together the extremes [0.2] when zeta is one [0.2] we've got 
the case of critical damping [0.2] that's [0.5] i know you all know that but i 
thought i'd write it down just for the sake of completeness [0.7] and in fact 
[0.3] when zeta [0.3] is greater than one [0.6] it's not really 
what we're looking at but when it's greater than one [0.3] we're off back to 
the overdamped case [14.1] and [0.7] again the extremes the other point was 
when [0.4] zeta is zero [1.1] that's these [0.2] there's natural frequency [0.
4] oscillations omega-N [0.3] and they just go on forever and ever and ever and 
[0.7] don't die down [12.4] and i think at this juncture [0.9] how's about we 
have [0.2] 'cause er this is a double header this morning for me isn't it [0.5] 
i think [0.4] 
sm0753: yeah [0.3] 
nm0724: hopefully yeah [0.4] i'm glad you're giving me support in that yeah [0.
6] so er [0.3] maybe er as i fancy making myself a cup of coffee [0.3] if we 
said five past [0.7] is that all right five past and we'll get going [0.4] and 
we'll get this wrapped up [0.3] so i'll see you all then have a wonderful time 
[0.3] in the fifteen minute break [0.5] or fourteen minutes