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<?xml version="1.0"?>

<!DOCTYPE TEI.2 SYSTEM "base.dtd">




<title>Critically damped response</title></titleStmt>

<publicationStmt><distributor>BASE and Oxford Text Archive</distributor>


<availability><p>The British Academic Spoken English (BASE) corpus was developed at the

Universities of Warwick and Reading, under the directorship of Hilary Nesi

(Centre for English Language Teacher Education, Warwick) and Paul Thompson

(Department of Applied Linguistics, Reading), with funding from BALEAP,

EURALEX, the British Academy and the Arts and Humanities Research Board. The

original recordings are held at the Universities of Warwick and Reading, and

at the Oxford Text Archive and may be consulted by bona fide researchers

upon written application to any of the holding bodies.

The BASE corpus is freely available to researchers who agree to the

following conditions:</p>

<p>1. The recordings and transcriptions should not be modified in any


<p>2. The recordings and transcriptions should be used for research purposes

only; they should not be reproduced in teaching materials</p>

<p>3. The recordings and transcriptions should not be reproduced in full for

a wider audience/readership, although researchers are free to quote short

passages of text (up to 200 running words from any given speech event)</p>

<p>4. The corpus developers should be informed of all presentations or

publications arising from analysis of the corpus</p><p>

Researchers should acknowledge their use of the corpus using the following

form of words:

The recordings and transcriptions used in this study come from the British

Academic Spoken English (BASE) corpus, which was developed at the

Universities of Warwick and Reading under the directorship of Hilary Nesi

(Warwick) and Paul Thompson (Reading). Corpus development was assisted by

funding from the Universities of Warwick and Reading, BALEAP, EURALEX, the

British Academy and the Arts and Humanities Research Board. </p></availability>




<recording dur="00:48:32" n="7490">


<respStmt><name>BASE team</name>



<langUsage><language id="en">English</language>

<language id="fr">French</language>



<person id="nm0724" role="main speaker" n="n" sex="m"><p>nm0724, main speaker, non-student, male</p></person>

<person id="om0725" role="observer" n="o" sex="m"><p>om0725, observer, observer, male</p></person>

<person id="sm0726" role="participant" n="s" sex="m"><p>sm0726, participant, student, male</p></person>

<person id="sm0727" role="participant" n="s" sex="m"><p>sm0727, participant, student, male</p></person>

<person id="sm0728" role="participant" n="s" sex="m"><p>sm0728, participant, student, male</p></person>

<person id="sm0729" role="participant" n="s" sex="m"><p>sm0729, participant, student, male</p></person>

<person id="sm0730" role="participant" n="s" sex="m"><p>sm0730, participant, student, male</p></person>

<person id="sm0731" role="participant" n="s" sex="m"><p>sm0731, participant, student, male</p></person>

<person id="sm0732" role="participant" n="s" sex="m"><p>sm0732, participant, student, male</p></person>

<person id="sm0733" role="participant" n="s" sex="m"><p>sm0733, participant, student, male</p></person>

<person id="sm0734" role="participant" n="s" sex="m"><p>sm0734, participant, student, male</p></person>

<person id="sm0735" role="participant" n="s" sex="m"><p>sm0735, participant, student, male</p></person>

<person id="sm0736" role="participant" n="s" sex="m"><p>sm0736, participant, student, male</p></person>

<person id="sm0737" role="participant" n="s" sex="m"><p>sm0737, participant, student, male</p></person>

<person id="sm0738" role="participant" n="s" sex="m"><p>sm0738, participant, student, male</p></person>

<person id="sm0739" role="participant" n="s" sex="m"><p>sm0739, participant, student, male</p></person>

<person id="sm0740" role="participant" n="s" sex="m"><p>sm0740, participant, student, male</p></person>

<person id="sm0741" role="participant" n="s" sex="m"><p>sm0741, participant, student, male</p></person>

<person id="su0742" role="participant" n="s" sex="u"><p>su0742, participant, student, unknown sex</p></person>

<person id="sm0743" role="participant" n="s" sex="m"><p>sm0743, participant, student, male</p></person>

<person id="sm0744" role="participant" n="s" sex="m"><p>sm0744, participant, student, male</p></person>

<person id="sm0745" role="participant" n="s" sex="m"><p>sm0745, participant, student, male</p></person>

<person id="sm0746" role="participant" n="s" sex="m"><p>sm0746, participant, student, male</p></person>

<person id="sm0747" role="participant" n="s" sex="m"><p>sm0747, participant, student, male</p></person>

<person id="sm0748" role="participant" n="s" sex="m"><p>sm0748, participant, student, male</p></person>

<person id="sm0749" role="participant" n="s" sex="m"><p>sm0749, participant, student, male</p></person>

<person id="sm0750" role="participant" n="s" sex="m"><p>sm0750, participant, student, male</p></person>

<person id="su0751" role="participant" n="s" sex="u"><p>su0751, participant, student, unknown sex</p></person>

<person id="sm0752" role="participant" n="s" sex="m"><p>sm0752, participant, student, male</p></person>

<person id="sm0753" role="participant" n="s" sex="m"><p>sm0753, participant, student, male</p></person>

<personGrp id="ss" role="audience" size="l"><p>ss, audience, large group </p></personGrp>

<personGrp id="sl" role="all" size="l"><p>sl, all, large group</p></personGrp>

<personGrp role="speakers" size="32"><p>number of speakers: 32</p></personGrp>





<item n="speechevent">Lecture</item>

<item n="acaddept">Cybernetics</item>

<item n="acaddiv">ps</item>

<item n="partlevel">UG1</item>

<item n="module">Systems</item>




<u who="nm0724"> that you know <pause dur="0.2"/> what's going down <pause dur="0.2"/> this morning <pause dur="0.5"/> # we got some guy where are you from anyway </u><pause dur="0.5"/> <u who="om0725" trans="pause"> Centre for Applied Language Studies </u><u who="nm0724" trans="latching"> there you are so he's local <pause dur="0.5"/> we # he so he's not that <pause dur="0.2"/> # oh you weren't here we had the Polish T-V lecture in # <pause dur="0.2"/> # <pause dur="0.7"/> lecture or videoing a lecture the other day <pause dur="0.4"/> but # <pause dur="0.3"/> that wasn't you lot have you lot been # some of you have been videoed haven't you <pause dur="1.0"/> the oh yeah when # some of you were in the lab i know others of you missed it but for # <pause dur="0.4"/> the N-B-C the American one <pause dur="0.5"/> # that went out on the # <pause dur="0.6"/> # N-B-C Today programme <pause dur="0.7"/> <trunc>s</trunc> <pause dur="0.3"/> oh no not B-B-C N-B-C <pause dur="0.2"/> so it was # <pause dur="0.3"/> i think something like thirteen-million Americans would have watched you # <pause dur="0.8"/> and there's been feedback saying how good you looked and things like that <pause dur="0.8"/> # <pause dur="0.2"/> and and at least one of you appeared on Sky T-V # or # filmed for Sky yesterday but <pause dur="0.2"/> oh <shift feature="voice" new="laugh"/> no names <shift feature="voice" new="normal"/> mentioned <pause dur="2.5"/> the celebrities in the audience it's amazing <pause dur="0.5"/> # i got a favour to ask in fact as well <pause dur="0.3"/> has anybody got <pause dur="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 <pause dur="0.4"/> has anybody got a video <pause dur="0.4"/> # machine or equipment or anything or have access to it <pause dur="1.8"/> you do <pause dur="1.1"/> well it's <trunc>n</trunc> <trunc>s</trunc> <trunc>n</trunc> <trunc>n</trunc> <trunc>n</trunc> <trunc>noth</trunc> nothing naughty i'm asking <vocal desc="laughter" n="ss" iterated="y" dur="2"/> for no <pause dur="0.5"/> # <pause dur="0.6"/> i i rushed out the house this morning and i forgot <pause dur="0.3"/> the <pause dur="0.2"/> Newsround tonight <pause dur="0.4"/> would it be possible <pause dur="0.2"/> for you </u><pause dur="0.5"/> <u who="sm0726" trans="pause"> yeah sure </u><pause dur="0.5"/> <u who="nm0724" trans="pause"> brill <pause dur="0.3"/> excellent <pause dur="0.4"/> 'cause </u><u who="sm0726" trans="overlap"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.8"/> <u who="nm0724" trans="pause"> # i think it starts four-<pause dur="0.2"/>fifty-five but i might <pause dur="0.4"/> but it's Beeb one <pause dur="0.3"/> four <kinesic desc="puts hand over mouth" iterated="n"/><gap reason="inaudible due to hand over mouth" extent="1 word"/><pause dur="0.2"/> something like that <pause dur="0.8"/> if it's poss just to video it <pause dur="0.2"/> i'll pay you handsomely <pause dur="0.2"/> you know # <pause dur="0.3"/> while this is being recorded <pause dur="0.4"/> i don't know what shall we say sort of ninety per cent in the exam something like that or <pause dur="0.4"/><vocal desc="laughter" n="ss" iterated="y" dur="1"/> that all right <pause dur="0.3"/> okay i mean # <trunc>al</trunc> all right ninety-five okay <pause dur="0.2"/> that's the final offer <pause dur="2.0"/><event desc="puts on transparency" iterated="n"/> just so this is this is me being # <pause dur="0.3"/> very generous <pause dur="0.2"/> so and <trunc>i</trunc> and if ever you say you said ninety-five and you don't get it <pause dur="0.3"/> then it's all been recorded <pause dur="0.3"/> so <pause dur="2.7"/><kinesic desc="turns on overhead projector showing transparency" iterated="n"/> right what it what it is on Newsround <pause dur="0.2"/> is oh don't panic don't panic

you've had all this before <pause dur="0.4"/> # what it is on Newsround is something to do with the Sony dog <pause dur="0.5"/> and they came here and <pause dur="0.5"/> we of course we're the most knowledgeable people on earth about the Sony dog <pause dur="0.4"/> Sony excepted <pause dur="0.2"/> and # <pause dur="0.3"/> they wanted a bit of waffle on it <pause dur="0.4"/> i think that's what it's about anyway it could be <pause dur="0.5"/> something completely different <pause dur="0.8"/> # <pause dur="1.0"/> let me let me remind you oh it's all right <pause dur="0.3"/> step this way step this way <pause dur="0.2"/> you need a crash helmet for this lecture <pause dur="0.8"/> it's # i know it's a bit of a <pause dur="0.6"/> a bit of a crash course <pause dur="0.3"/> the the <trunc>s</trunc> <pause dur="0.2"/> stuff we were doing <pause dur="0.2"/> if you remember <pause dur="0.4"/> we have a system that we're dealing with <pause dur="0.3"/> and we're shoving a step input into it <pause dur="0.4"/> step input is switching it on <pause dur="0.2"/> <shift feature="pitch" new="high"/> bing<shift feature="pitch" new="normal"/> step input <pause dur="0.4"/> the <trunc>pra</trunc> this is practical # <vocal desc="laughter" iterated="y" dur="1"/> experimentation <pause dur="0.6"/> and # <pause dur="0.3"/> we're seeing how the system responds and we looked at the first type of system <pause dur="0.3"/> where it's overdamped so <pause dur="0.3"/> the system responds the output <pause dur="0.4"/> morning <pause dur="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 <pause dur="1.4"/> we then had a look <pause dur="0.4"/> <trunc>thi</trunc> this is hopefully how i <pause dur="0.2"/> how i left you last time we were on <pause dur="0.4"/> # was looking at the critically damped case <pause dur="0.6"/> and how far did we go fine <pause dur="0.2"/> so this this you've had a look at this bit so far so if i go over this you won't <pause dur="0.2"/> or shouldn't be saying what the hell is that <pause dur="0.3"/> it should be yeah i remember that way back <pause dur="0.2"/> beyond the <pause dur="0.4"/> days gone by <pause dur="0.4"/> the <pause dur="0.2"/> in the <pause dur="0.3"/> critically damped case just to remind you the critically damped case <pause dur="0.4"/> is really this sort of <pause dur="0.3"/> mathematical ideal on the cusp <pause dur="0.4"/> probably never really happens in practice type <pause dur="0.2"/> it it's a it's an ideal type of thing to get to <pause dur="0.5"/> it's sort of # <pause dur="0.3"/> in a political sense it's the pure communist state you know it's an <trunc>ol</trunc> <trunc>i</trunc> ideal but it's never # going to actually occur in practice <pause dur="0.6"/> # <pause dur="0.6"/> here we got a system <pause dur="0.2"/> output <pause dur="0.3"/> the the system is G input is U which is a step input <pause dur="0.4"/> and where we'd got to <pause dur="0.4"/> was <pause dur="0.2"/> simply at the point of looking at what the output response is <pause dur="0.8"/> to a step input <pause dur="1.0"/> and # <pause dur="0.5"/> # again

what we'd done before well what what i'd done and some of you had listened in <pause dur="0.5"/> was to look at the sort of extreme cases <pause dur="0.4"/> when <trunc>thi</trunc> this is what it looks like the output looks like in terms of time physically what the output looks like <pause dur="0.5"/> and there are three <pause dur="0.2"/> values to it or three parts to it <pause dur="0.6"/> one is the just the gain K is just a number <pause dur="0.4"/> alpha <pause dur="0.2"/> apart from being a Greek symbol just to make it look more difficult <pause dur="0.3"/> is also just a number <pause dur="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 <pause dur="0.5"/> we've then got one <pause dur="0.3"/> now this this effectively will be the steady state value when things have settled down <pause dur="0.3"/> and all we've got are two exponential terms <pause dur="0.7"/> which we can look at at the extreme cases <pause dur="0.2"/> and the extreme cases being <pause dur="0.2"/> when we switched the thing on <pause dur="0.4"/> and <pause dur="0.2"/> it's like # time is zero <pause dur="0.2"/> it's the big bang time <pause dur="0.3"/> well in a physical sense <pause dur="0.2"/> that is not in a <pause dur="0.3"/> # well any other sense <pause dur="0.4"/> and i've got to watch what i say here haven't i 'cause <pause dur="0.2"/> i mean <trunc>th</trunc> this is

this is with language studies you get sort of <pause dur="0.2"/> # mystical foreign students # <pause dur="0.5"/> listening in on this or <pause dur="0.3"/> yeah </u><pause dur="0.6"/> <u who="om0725" trans="pause"> possibly </u><pause dur="0.2"/> <u who="nm0724" trans="pause"> okay <pause dur="0.3"/> for the so for the <pause dur="0.3"/> the Latin American students that might listen to this transcript <pause dur="0.3"/> the big bang i'm talking purely in a <pause dur="0.2"/> sort of Stephen Hawking sense not in # <pause dur="0.4"/> # other <pause dur="0.5"/> other connotations <pause dur="0.5"/> and # <pause dur="0.4"/> anyway so actually when when <trunc>t</trunc> <vocal desc="laugh" iterated="n"/> oh God let's go for it <pause dur="0.2"/> when T is zero <pause dur="0.4"/> then exponential of zero is </u><pause dur="0.9"/><u who="sm0727" trans="pause"> one </u><pause dur="0.3"/><u who="nm0724" trans="pause"> one great <pause dur="0.4"/> and when <pause dur="0.5"/> this lot is zero because we've got a T in there <pause dur="0.2"/> that going to zero the whole lot <pause dur="0.5"/> of that goes to zero <pause dur="0.3"/> so all we've got when we switch the thing on <pause dur="0.3"/> this is when we apply the step input at that time we go <shift feature="pitch" new="high"/> zing <shift feature="pitch" new="normal"/> <pause dur="0.3"/> then the output <pause dur="0.4"/> will be this value <pause dur="0.2"/> one-minus-one <pause dur="0.4"/> and one-minus-one is </u><pause dur="0.9"/> <u who="sm0728" trans="pause"> zero </u><pause dur="0.8"/> <u who="nm0724" trans="pause"> one student knows anyway that # <trunc>th</trunc> <pause dur="0.2"/> yeah one-minus-one is zero <pause dur="0.2"/> which means that the output <pause dur="0.2"/> is zero <pause dur="0.4"/> which is pretty obvious we switch the thing on and the output's # well i was going to say <pause dur="0.2"/> i was going to swear but i won't <pause dur="0.4"/> # the output is zero <pause dur="0.5"/> # but why not <pause dur="0.4"/> but # now let's let's take the other case <pause dur="0.2"/>

when time goes off to a big number <pause dur="0.3"/> such as infinity or the end of the lecture whichever is first <pause dur="0.5"/> and <pause dur="0.2"/> in this case the exponential minus infinity <pause dur="0.4"/> is </u><pause dur="2.0"/> <u who="sm0729" trans="pause"> must be zero </u><pause dur="0.3"/> <u who="nm0724" trans="pause"> must be zero <pause dur="0.2"/> it sounds good doesn't it it sounds good <pause dur="0.3"/> so that must be zero <pause dur="0.4"/> and here we've got <pause dur="0.4"/> # <pause dur="0.2"/> time is off to infinity or a big number <pause dur="0.9"/> and this bit is down to to zero <pause dur="0.3"/> so what's <pause dur="0.3"/> # <pause dur="0.3"/> anything multiplied by infinity is </u><pause dur="0.5"/> <u who="sm0730" trans="pause"> infinity </u><u who="nm0724" trans="latching"> infinity <pause dur="0.3"/> except for one specific case because anything multiplied by zero is </u><pause dur="0.3"/><u who="sm0731" trans="pause"> zero </u><pause dur="0.2"/> <u who="nm0724" trans="pause"> zero <pause dur="0.4"/> and # we're on one of these sort of <pause dur="0.5"/> <trunc>parad</trunc> what is it a <trunc>p</trunc> a paradigm or something i don't know what <pause dur="0.2"/> or is that some exotic drink </u><pause dur="0.7"/> <u who="sm0732" trans="pause"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.3"/> <u who="nm0724" trans="pause"> no all right anyway <pause dur="0.2"/> it's zero multiplied by any old thing and T is a big number <pause dur="0.3"/> so the whole lot's going to go to zero there <pause dur="0.3"/> so it's because <pause dur="0.2"/> that's <pause dur="0.6"/> zip all <pause dur="0.4"/> so we've got one-minus-zero <pause dur="0.2"/> which is </u><pause dur="0.8"/> <u who="sm0733" trans="pause"> one </u><pause dur="0.3"/> <u who="nm0724" trans="pause"> one thanks very much <pause dur="0.5"/> well we <trunc>ge</trunc> we're getting there even if it's the odd sort of person there <pause dur="0.3"/> so in a steady state <pause dur="0.2"/> this would be <pause dur="0.3"/> K-over-alpha-squared whatever the number is <pause dur="0.2"/> multiplied by one <pause dur="0.3"/> which turns out to be <pause dur="0.2"/>

K-over-alpha-squared <pause dur="0.8"/> so that's the critically damped case <pause dur="0.4"/> <kinesic desc="changes transparency" iterated="y" dur="3"/><vocal desc="clears throat" iterated="n"/><pause dur="0.3"/> that's the steady state value <pause dur="1.5"/> and the two extremes <pause dur="0.4"/> # <pause dur="0.4"/> are the ones we've just looked at <pause dur="0.3"/> the steady state when time goes off to infinity <pause dur="0.6"/> or a or a big number <pause dur="0.3"/> whichever you want <pause dur="0.8"/> # is this number K-over-alpha-squared which is the <pause dur="0.3"/> the inherent <pause dur="0.3"/> gain <pause dur="0.2"/> of the system <pause dur="2.4"/> again they're just numbers <pause dur="0.3"/> K and alpha <pause dur="0.3"/> are just numbers <pause dur="1.0"/> i mean Kay is also a girl's name i know that <pause dur="0.2"/> but for the purposes of this thing <pause dur="0.6"/> just treat her as a number <pause dur="0.3"/> if anybody knows <pause dur="0.7"/> anybody called Kay <pause dur="2.3"/> tonight <pause dur="0.4"/> say to her Kay you're just a number <pause dur="4.2"/> if if she gets upset about that <pause dur="0.7"/> you know especially if you say you're number fifty-eight or something like that <pause dur="0.2"/> she gets upset then say no the prof says you're just a number <pause dur="1.3"/> say how does he know <pause dur="0.7"/> # <pause dur="1.5"/> and then in the initial <pause dur="1.2"/> does anybody know anybody called Kay just to get you into trouble here <pause dur="0.6"/> just as well <pause dur="0.2"/> just as well </u><pause dur="0.5"/> <u who="sm0734" trans="pause"> bloke </u><u who="nm0724" trans="overlap"> don't <pause dur="0.4"/> pardon </u>

<pause dur="0.6"/> <u who="sm0734" trans="pause"> there's a bloke called Kay </u><pause dur="0.5"/> <u who="nm0724" trans="pause"> is there <pause dur="1.1"/> # <pause dur="0.2"/> okay <pause dur="0.4"/> well i'll <vocal desc="laughter" iterated="y" dur="1"/></u><u who="sm0735" trans="overlap"> <gap reason="inaudible" extent="2 secs"/> </u><u who="nm0724" trans="overlap"> right <vocal desc="clears throat" iterated="y" dur="1"/> <shift feature="pitch" new="low"/> yeah well <vocal desc="laughter" n="ss" iterated="y" dur="2"/> the # <shift feature="pitch" new="normal"/> </u><pause dur="0.3"/> <u who="sm0736" trans="pause"> these Japanese <gap reason="inaudible" extent="2 secs"/> </u><pause dur="0.5"/> <u who="nm0724" trans="pause"> really <pause dur="2.6"/> <shift feature="pitch" new="low"/> okay let's move on <shift feature="pitch" new="normal"/><pause dur="1.9"/><vocal desc="laughter" iterated="y" dur="1"/> # the in fact <pause dur="0.2"/> not that i'm going to go through 'cause <pause dur="0.2"/> you know we got language students who don't know what all this derivative stuff is <pause dur="0.3"/> but in fact the derivative <pause dur="0.2"/> # of the output <pause dur="0.3"/> when T is zero <pause dur="0.4"/> also <pause dur="0.4"/> turns out to be zero <pause dur="0.2"/> you know we had that we had that little trinket <pause dur="0.2"/> cropping up before and here it comes again <pause dur="0.2"/> it's not particularly important no big deal <pause dur="0.5"/> all it means is when <pause dur="0.4"/> you # <pause dur="0.3"/> # # it's it's very much like <pause dur="0.5"/> # <pause dur="0.4"/> # a member of the opposite sex <pause dur="0.2"/> when you actually suggest something to them <pause dur="0.2"/> it takes them a bit of time to get going <pause dur="0.2"/> you can't suggest something to them and they're off straight away <pause dur="0.2"/> and the same with this system <pause dur="0.2"/> so in fact the <pause dur="0.2"/> the other person could be a critically damped system <pause dur="0.7"/> # <pause dur="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 <pause dur="0.5"/> what this critically damped <pause dur="0.4"/> system looks like <pause dur="0.4"/> there's no overshoot <pause dur="0.3"/>

the dotted this this dotted thing here <pause dur="0.3"/> is the steady state value <pause dur="0.3"/> # this is the output <pause dur="0.5"/> here's time along here <pause dur="0.3"/> we switch the switch <pause dur="0.3"/> and up it goes <pause dur="0.3"/> but it doesn't actually <pause dur="1.3"/> overshoot <pause dur="0.4"/> in theory <pause dur="0.2"/> it doesn't overshoot the final <pause dur="0.2"/> value that it settles down to <pause dur="2.3"/>

in fact it is the most rapid <pause dur="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 <pause dur="0.6"/> without <pause dur="0.6"/> any overshoot <pause dur="3.8"/> for the overdamped case you remember <pause dur="0.3"/> it <pause dur="0.2"/> it <pause dur="0.2"/> didn't overshoot it # <pause dur="0.2"/> came in from <unclear>once i</unclear> eventually got there <pause dur="2.6"/> and i've said here it's desirable this particular <pause dur="0.3"/> goal this particular type critically damped response <pause dur="0.4"/> if you're looking at machine tools <pause dur="0.5"/> # or robots <pause dur="0.2"/> in fact or # a lot of production machinery <pause dur="0.2"/> welding machines is a a prime example <pause dur="0.7"/> i know tonight you'll all go home and play with a welding machine <pause dur="0.5"/> as you do <pause dur="0.3"/> and with that welding machine the aim the target <pause dur="0.4"/> is to make the thing critically damped <pause dur="0.5"/> so <pause dur="0.6"/> if if you can imagine for a moment <pause dur="0.2"/> that i am a welding machine <pause dur="0.4"/> and <pause dur="0.3"/> this the top of the projector is the thing <pause dur="0.2"/> like what i am supposed

to be welding <pause dur="0.4"/> then <pause dur="0.2"/> here's my little torch <pause dur="0.2"/> or stick or whatever the welding thing is <pause dur="0.4"/> and i the design of it <pause dur="0.2"/> i want this arm to come in as quickly as possible <pause dur="0.7"/> do do you need a <trunc>v</trunc> a vocal commentary <pause dur="0.4"/><event desc="moves microphone closer to mouth" iterated="y" dur="2"/> he moved his arm <vocal desc="laughter" n="ss" iterated="y" dur="2"/> in the direction of <vocal desc="laughter" n="sl" iterated="y" dur="1"/> so i want the arm to come in as quick as possible <pause dur="0.2"/> to to that point <shift feature="pitch" new="high"/> zing <shift feature="pitch" new="normal"/> like that do the <vocal desc="buzzing noise" iterated="y" dur="1"/> do the welding and move away again <pause dur="0.3"/> now if it's overdamped the case we saw before <pause dur="0.2"/> it will not come in zing it will come in <shift feature="tempo" new="ll"/> zing <shift feature="tempo" new="normal"/> and eventually get there <pause dur="0.6"/> if it's as the case we're going to move on to <pause dur="0.2"/> it goes past it overshoots <pause dur="0.3"/> then of course it would go <pause dur="0.3"/> zing and we messed the <pause dur="0.2"/> and you can imagine whatever it is we've gone too far <pause dur="0.5"/> and know what going too far gets you <pause dur="1.0"/> # so this critically damped case is the <pause dur="0.3"/> the <pause dur="0.2"/> the perfect the ideal case if we had <pause dur="0.3"/> a perfect welder <pause dur="0.4"/> see knows all about perfect welders he can afford to miss the first ten minutes <pause dur="0.4"/> if we had the perfect welder <pause dur="0.3"/> it would be critically damped <pause dur="0.3"/> however <pause dur="0.5"/> i this this thing here <pause dur="0.5"/> is important <pause dur="0.6"/> that

that's really in theory <pause dur="0.2"/><kinesic desc="indicates point on transparency" iterated="n"/> this is the theoretical <pause dur="0.2"/> for Christ's sake if you get a job <pause dur="0.3"/> at the end of this course <pause dur="0.3"/> as a <pause dur="0.2"/> welding machine setter upper <pause dur="0.3"/> or whatever it happens to be <pause dur="0.2"/> or someone who's in charge of people who set welding machines up or <pause dur="0.2"/> or whatever <pause dur="0.4"/> # then <pause dur="0.8"/> don't <pause dur="0.3"/> do it <pause dur="0.2"/> quite this way <pause dur="0.3"/> aim for this as a perfect thing but remember <pause dur="0.4"/> probably you're going to get a bit of noise floating around probably you're going to get a bit of variability <pause dur="0.4"/> one <pause dur="0.2"/> robot's welding stick <pause dur="0.2"/> might be a little bit longer than another robot's welding stick <pause dur="0.3"/> you know in this world <pause dur="0.2"/> robots have welding sticks of different lengths we have to be # <pause dur="0.4"/> aware of that <pause dur="0.4"/> and that might mean if you've set the thing up <pause dur="0.5"/> to come in just from one side the the critically damped case <pause dur="0.5"/> it <pause dur="0.2"/> it might <pause dur="0.3"/> bang the <pause dur="0.3"/> so you might break it it might <pause dur="0.2"/> you you could go too far <pause dur="0.4"/> # the other thing with noise <pause dur="0.2"/> really you may not want this to overshoot and # with a bit of noise on the top <pause dur="0.3"/> the <trunc>si</trunc> the signal wouldn't look like that it would come in <pause dur="0.3"/>

and it it might go like which means you would in fact be <pause dur="0.3"/> overshooting a little bit <pause dur="1.0"/> so in practice what happens is <pause dur="0.9"/> # you would slug it down <pause dur="0.3"/> a bit <pause dur="0.3"/> so in <pause dur="0.6"/> a a lot of manufacturing cases in fact it's really slugged down a hell of a lot <pause dur="0.5"/> # <pause dur="0.4"/> typical production line down at <pause dur="0.3"/> say Ford <pause dur="0.2"/> who are <pause dur="0.3"/> about to <pause dur="0.7"/> close the production line down i don't know whether that's any coincidence <pause dur="0.5"/> but so at Dagenham the robots there they have some big KUKA robots that do welding and parts manipulation and so on <pause dur="0.4"/> and they're slowed down to about a third <pause dur="0.3"/> sort of three times if you like <pause dur="0.2"/> slower than they could actually <pause dur="0.4"/> operate at <pause dur="0.3"/> for safety reasons for that <pause dur="0.2"/> reason that if they operated faster than that <pause dur="0.5"/> then <pause dur="0.3"/> they could the the to achieve <pause dur="0.3"/> more production because if the things are operating faster they get more things going through more welds three welds a <trunc>m</trunc> a minute instead of one weld or whatever it happens to be <pause dur="0.5"/> but as you get towards that optimum this critically damped case which is what we're looking at <pause dur="0.5"/> you've got

all sorts of potential problems because of the variability in what you're dealing with <pause dur="0.5"/> so this is a an ideal a theoretical case to aim for <pause dur="0.3"/> here's the theory <pause dur="0.3"/> now let's detune slow things down let's come back to the overdamped case a bit in a practical way <pause dur="0.4"/> in order that things don't blow up and we don't break things and so on <pause dur="0.2"/> so forth <pause dur="0.3"/> so that's what we're about <pause dur="1.4"/><kinesic desc="changes transparency" iterated="y" dur="14"/> so there # critically damped is a sort of <pause dur="0.3"/> theoretical ideal <pause dur="0.8"/> # it's a bit like my wage packet it's completely theoretical but the <pause dur="0.9"/><vocal desc="clears throat" iterated="y" dur="1"/> it's not ideal either <pause dur="1.3"/><vocal desc="clears throat" iterated="n"/> it seems seem that the language <pause dur="0.2"/> they get paid loads of dosh over there so <pause dur="0.8"/> they get <trunc>p</trunc> <pause dur="0.3"/> get paid it all in foreign money and # <pause dur="0.2"/><vocal desc="laughter" n="sl" iterated="y" dur="3"/> <pause dur="0.6"/> right <pause dur="0.6"/> # we've done <pause dur="0.6"/> the overdamped case <pause dur="0.5"/> we've done the critical damped <pause dur="0.2"/> case <pause dur="1.1"/> and # <pause dur="0.4"/> we're going to have a look at the <pause dur="0.3"/> this is this is in fact this is a really sad moment <pause dur="0.4"/> because this underdamped response <pause dur="0.5"/> this is it this is the final thing we're going to be covering on the course <pause dur="0.4"/> so <pause dur="0.6"/> i know i know it's a time for holding back the tears for <pause dur="0.7"/> # you know <pause dur="0.8"/>

stifling <pause dur="0.4"/> not a yawn <pause dur="0.2"/> but <pause dur="0.8"/> stifling something or other <pause dur="0.9"/> # so the <trunc>o</trunc> the the overdamped <pause dur="0.2"/> it didn't didn't overshoot critically damped's right on the cusp <pause dur="0.4"/> underdamped is the the final thing we're going to look at <pause dur="0.2"/> and it's called underdamped because it's underdamped <pause dur="0.5"/> # <pause dur="0.3"/> the the damping <pause dur="0.3"/> is not enough <pause dur="0.4"/> to stop <pause dur="0.3"/> overshoots <pause dur="0.3"/> essentially <pause dur="1.0"/> and this this type of response the output <pause dur="0.4"/> overshoots its steady state value <pause dur="0.4"/><kinesic desc="changes transparency" iterated="y" dur="5"/> so i'll just <pause dur="0.2"/> flick back <pause dur="0.4"/> to this <pause dur="0.2"/> previous overhead <pause dur="1.1"/> he put the previous overhead on the projector <pause dur="0.5"/> flick back to that <pause dur="0.3"/> then in this case we was the the sort of ideal in the <trunc>n</trunc> furthest we could go <pause dur="0.2"/> quickest we could go without actually overshooting here <pause dur="0.2"/> we're actually going to go past this final value <pause dur="0.3"/> or <pause dur="0.3"/> # for language students from <pause dur="0.3"/> # the southern English <pause dur="0.3"/> past the final value <pause dur="1.1"/> # <pause dur="0.2"/> the transfer function roots in this case you remember we had <pause dur="0.5"/> the first case the overdamped case the roots of the denominator <pause dur="0.5"/> were <pause dur="0.3"/> were like two <pause dur="0.2"/> different <pause dur="0.2"/> roots as it were but real <pause dur="0.2"/> roots <pause dur="0.7"/> and in the <pause dur="0.6"/>

critically damped case <pause dur="0.3"/> the roots were in fact exactly the same <pause dur="2.6"/> and in this case <pause dur="0.3"/> for the underdamped case <pause dur="0.3"/> we've got <pause dur="0.3"/> complex conjugates <pause dur="0.2"/> we've got # <pause dur="0.6"/> # <pause dur="0.2"/> a real part and an imaginary part <pause dur="5.3"/> and and in this case for the underdamped response <pause dur="0.4"/> the output <pause dur="0.2"/> oscillates <pause dur="1.0"/> following a step input <pause dur="2.5"/> oscillates meaning it it goes backwards and forwards <pause dur="3.6"/> and just <pause dur="0.3"/> just to see what i'm getting at here what we're talking about here <pause dur="0.5"/> an an experiment for tonight is <pause dur="0.3"/> trying to get the <pause dur="0.2"/> key into the door of your room or house wherever you're living <pause dur="0.7"/> # <pause dur="0.3"/> for a <pause dur="0.3"/> an overdamped case just just go normally <pause dur="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 <pause dur="0.8"/> just for the purposes of experimentation <pause dur="0.2"/> have a few drinks <pause dur="0.4"/> and then try <pause dur="0.2"/> and get the key in the door <pause dur="0.2"/> and you'll find <pause dur="0.2"/> more than like well i find anyway i <trunc>don</trunc> <pause dur="0.3"/> is is that the key you'll have to sort of <trunc>j</trunc> <trunc>j</trunc> jiggle it around a bit like that <pause dur="0.3"/> and eventually you get

the key in the door <pause dur="0.3"/> the more drinks the have <pause dur="0.2"/> you have the more jiggling around that you have to do <pause dur="0.3"/> until if you have <pause dur="0.3"/> too many i'm not suggesting you have too many but if you have too many <pause dur="0.3"/> probably goes completely unstable and you'll fall over sideways while you're trying to <vocal desc="laughter" n="ss" iterated="y" dur="1"/> get <pause dur="0.2"/> the key in the door <pause dur="0.6"/> but this one the oscillations are they're trying to to get <trunc>r</trunc> right on to the <unclear>then you over</unclear> you go past and you come you know the <pause dur="0.2"/> you know the feeling you go <pause dur="0.2"/> it's a <pause dur="0.2"/> it's a bit like my wife's driving so <pause dur="0.9"/><vocal desc="laughter" n="ss" iterated="y" dur="1"/> <vocal desc="cough" iterated="n"/><pause dur="0.7"/> don't laugh it's true <trunc>unfort</trunc> <vocal desc="laugh" iterated="n"/><pause dur="1.0"/> that's another another rear reflector i shall be mending tomorrow <pause dur="1.7"/> # the amount of damping <pause dur="0.8"/> dictates <pause dur="0.3"/> how quickly these oscillations die down <pause dur="1.1"/> a lot of damping <pause dur="0.2"/> the oscillations don't hang around for very long <pause dur="3.3"/> not much damping and they stick around for quite a while <pause dur="5.7"/> and in the limits <pause dur="0.4"/> if there was no damping <pause dur="2.2"/> then <pause dur="0.3"/> it's <pause dur="0.2"/> purely <pause dur="0.2"/> it's just oscillating <pause dur="0.5"/> off it goes <pause dur="1.7"/> just oscillates <pause dur="3.6"/> and for the foreign language students <pause dur="0.3"/> very much like a Citroen Diane suspension <pause dur="1.0"/>

it's getting <pause dur="0.2"/> towards that <pause dur="5.5"/> or <distinct lang="fr">le Diane de Citroen</distinct> <pause dur="0.2"/> or whatever it <vocal desc="laughter" iterated="y" dur="1"/> <pause dur="1.6"/> whatever it is <pause dur="1.0"/> am i supposed to interact with this or should this be a </u><pause dur="0.3"/> <u who="om0725" trans="pause"> <gap reason="inaudible" extent="1 sec"/> </u><u who="nm0724" trans="latching"> <vocal desc="laughter" iterated="y" dur="2"/> <pause dur="0.4"/> just that no i mean normally this is deadly serious is these lectures <pause dur="0.3"/> are they not <vocal desc="laughter" n="ss" iterated="y" dur="1"/> <pause dur="0.2"/> and # </u><pause dur="0.3"/> <u who="sm0737" trans="pause"> compared with your other ones </u><pause dur="0.2"/> <u who="nm0724" trans="pause"> compared with other ones yeah </u><u who="sm0738" trans="overlap"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.3"/> <u who="nm0724" trans="pause"> that's right so this is i mean there have been no Skoda jokes today at all <pause dur="1.8"/> they can't remember any </u><u who="sm0739" trans="overlap"> <gap reason="inaudible" extent="1 sec"/> opportunities for one </u><pause dur="1.1"/> <u who="nm0724" trans="pause"> oh i don't need opportunities <gap reason="inaudible" extent="1 sec"/> </u><pause dur="1.3"/> <u who="sm0740" trans="pause"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.2"/> <u who="nm0724" trans="pause"> do you want a Skoda joke </u><pause dur="0.9"/> <u who="sm0741" trans="pause"> <gap reason="inaudible" extent="1 sec"/></u><u who="nm0724" trans="latching"> yeah <pause dur="0.6"/> i don't know have i had any Skoda this is ah this is for the <pause dur="0.2"/> Czech <pause dur="0.2"/> listeners # <pause dur="1.2"/> my my wife is Czech this is a true story <pause dur="0.2"/> actually <pause dur="0.8"/> and # <pause dur="0.9"/> <trunc>w</trunc> years ago when she first came to England <pause dur="1.0"/> # <pause dur="0.4"/> the <pause dur="0.2"/> university photographer caught us at the degree ceremony <pause dur="0.3"/> and said aha you're Czech i've got a Skoda joke as people do <pause dur="0.2"/> tend to do <pause dur="0.7"/> and is <trunc>i</trunc> you may have heard this bit but stick with it <pause dur="0.4"/> and he said # <pause dur="0.4"/> why why do Skodas have a heated rear windscreen <pause dur="1.0"/> and my wife # <pause dur="0.4"/> and he said so you can keep your hands

warm when you're pushing it <pause dur="1.3"/><vocal desc="laughter" n="ss" iterated="y" dur="1"/> and <pause dur="0.3"/> we we sort of <pause dur="0.4"/> well me and another guy we were sort of <pause dur="0.2"/> laughing a little bit <pause dur="0.2"/> 'cause we haven't heard it before see it was many years ago <pause dur="0.7"/> but my wife didn't laugh at all <pause dur="0.7"/> i thought well maybe he's offended her <trunc>wh</trunc> <trunc>wh</trunc> why aren't you laughing it's quite funny <pause dur="0.4"/> and she said well Skodas don't have a heated rear windscreen <pause dur="1.3"/><vocal desc="laughter" n="ss" iterated="y" dur="2"/> <vocal desc="laughter" n="sl" iterated="y" dur="3"/> <pause dur="1.1"/> right <pause dur="0.3"/> way to go <pause dur="0.2"/><vocal desc="laughter" iterated="y" dur="1"/> <pause dur="1.0"/> the true jokes are the funniest <pause dur="0.7"/> # as the damping is <trunc>in</trunc> sorry that's got nothing to do with the lecture but what the hell <pause dur="0.5"/> as the damping is increased so we get <pause dur="0.4"/> to the critically damped case <pause dur="0.2"/> the case <pause dur="0.2"/> like what we were at <pause dur="0.3"/> before <pause dur="1.3"/><vocal desc="clears throat" iterated="n"/> so no damping it's just <vocal desc="buzzing noise" iterated="y" dur="1"/><pause dur="0.3"/> purely oscillatory <pause dur="0.2"/> increase the damping increase the damping <pause dur="0.3"/> and we will get <pause dur="0.2"/> the critically damped case where you don't get any overshoots <pause dur="0.3"/> no oscillations at all <pause dur="2.9"/> i just put the what a shame 'cause that's what i felt like <pause dur="0.7"/> that's an optional extra as far as your notes are concerned <pause dur="0.4"/> if anybody in the exam <pause dur="0.3"/> puts something like what a shame <pause dur="0.3"/> then well <pause dur="0.2"/> it's up

to you isn't it <pause dur="1.9"/> the decision is yours <pause dur="0.3"/><vocal desc="clears throat"/><pause dur="2.5"/><vocal desc="sneeze" n="su0742" iterated="n"/> bless you <pause dur="4.1"/> has anybody seen Bicentennial Man <pause dur="0.7"/> yet <pause dur="2.1"/> no <pause dur="2.0"/> okay <pause dur="1.1"/> so <trunc>y</trunc> <pause dur="0.3"/><event desc="moves microphone closer to mouth" iterated="y" dur="7"/> note we are with science students engineering students <pause dur="0.2"/> we're not into arty things like <pause dur="0.4"/> Robin Williams <pause dur="0.2"/> films and stuff so has <trunc>every</trunc> everybody finished with that </u><pause dur="0.4"/> <u who="sm0743" trans="pause"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.6"/> <u who="nm0724" trans="pause"> okay <pause dur="1.6"/> <event desc="moves microphone closer to mouth" iterated="y" dur="2"/> we wait while the slow student catches up <vocal desc="laughter" n="sl" iterated="y" dur="4"/> <pause dur="0.5"/> well apparently it's about <pause dur="0.2"/> some i i only know this 'cause some Argentinian journalist contacted me to say what do you think of it <pause dur="0.4"/> as Argentinian journalists do </u> <pause dur="0.3"/> <u who="sm0744" trans="pause"> based on the Asimov story </u><pause dur="1.0"/> <u who="nm0724" trans="pause"> i think it's yeah it <gap reason="inaudible" extent="1 sec"/> this robot </u><u who="sm0745" trans="overlap"> about the robot that <unclear>wanted to be</unclear> human </u><u who="nm0724" trans="overlap"> <trunc>th</trunc> <pause dur="0.3"/> that's right it couldn't be human and wants to have human feelings and stuff like that <pause dur="0.5"/> so the journalist asked me <pause dur="0.3"/> can robots have <pause dur="0.6"/> feelings and so on <pause dur="1.2"/><vocal desc="clears throat" iterated="n"/><pause dur="0.2"/> that's why i've put what a shame that was the link putting what a shame 'cause the system doesn't <pause dur="0.5"/> have <pause dur="0.2"/> oscillations <pause dur="3.8"/><vocal desc="laughter" iterated="y" dur="5"/><kinesic desc="changes transparency" iterated="y" dur="2"/> that's it <pause dur="0.9"/> let's go for the next one <pause dur="0.3"/> now unfortunately <pause dur="0.2"/> you're not going to get away with this lightly <pause dur="0.3"/> i mean normally because we don't do stuff like maths and

things like that i mean <pause dur="0.2"/> but we have to because we've got <pause dur="0.4"/> # language student listening in <pause dur="0.2"/> we have to do some tough stuff <pause dur="0.2"/> okay <pause dur="0.5"/> so # <pause dur="0.3"/> purely for this purpose we'll <pause dur="0.2"/> throw some equations and maths on the board just so it looks <pause dur="0.2"/> as though we actually do some work <pause dur="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 <pause dur="0.8"/> <trunc>wh</trunc> how the hell do they know all that stuff <pause dur="0.8"/> and then once once we're not recording then we'll get back to the easy life that we normally <pause dur="0.8"/> do <pause dur="0.5"/><vocal desc="laughter" iterated="y" dur="1"/> <pause dur="0.4"/> like

Skoda jokes and things like that <pause dur="0.5"/> right <pause dur="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 <pause dur="0.4"/> and in this <pause dur="0.5"/> instance <pause dur="0.2"/> we have got <pause dur="0.3"/> complex conjugate roots <pause dur="0.2"/><vocal desc="clears throat" iterated="y" dur="1"/><pause dur="0.3"/> so the <pause dur="0.3"/> the transfer function G <pause dur="0.3"/> that relates input to output <pause dur="0.9"/> has this gain K <pause dur="0.2"/> there she is again on the top line <pause dur="0.2"/> or he again <pause dur="0.2"/> <shift feature="pitch" new="low"/> or he <shift feature="pitch" new="normal"/> again <pause dur="0.4"/> and on the <pause dur="0.3"/><shift feature="pitch" new="low"/> and on the bottom line <pause dur="0.3"/><shift feature="pitch" new="normal"/> we have <pause dur="0.6"/> S-plus-alpha-minus-J-beta <pause dur="0.9"/> bring in another Greek symbol <pause dur="0.2"/> make it more complicated <pause dur="0.6"/> for the Greek listeners this is probably all right though <pause dur="0.5"/> and and multiply by S-plus-alpha-<pause dur="0.2"/>plus-<pause dur="0.2"/>J-beta <pause dur="2.5"/> and you can see <pause dur="0.2"/> here why they use the term J <pause dur="0.3"/> and not <pause dur="0.3"/> some word like egg <pause dur="0.4"/> because if they called J egg <pause dur="0.3"/> then we'd have egg beta <pause dur="0.2"/> and that would be a bit silly <vocal desc="laughter" iterated="y" n="sl" dur="4"/> <pause dur="0.4"/> see there's there's logic in these symbols you thought why they use J now you know <pause dur="0.2"/> now you know <pause dur="0.8"/> # <pause dur="0.4"/> so we've got complex conjugate routes <pause dur="0.2"/> i'll just <pause dur="0.2"/> just to make it look again for the language people listening in <pause dur="0.2"/>

make it multiply things out and so on <pause dur="0.4"/> and we can also get <pause dur="0.3"/> if we multiply this stuff out to get rid of the J <pause dur="1.5"/> or egg if you like whichever the alternative <pause dur="0.8"/> # <pause dur="0.3"/> just i thought i'd <trunc>scramb</trunc> scramble <shift feature="voice" new="laugh"/> it up oh God <vocal desc="laughter" iterated="y" dur="1"/><shift feature="voice" new="normal"/> no no no that's enough <pause dur="0.3"/> that was almost a joke there <pause dur="0.8"/> so scramble it up <pause dur="0.2"/> and this is just looking at the denominator <pause dur="0.5"/> forget K's forgotten for the moment <pause dur="1.3"/> poor girl <pause dur="1.0"/> # and if we can also if we multiply it out <pause dur="0.3"/> because this this little bit this little <pause dur="0.5"/> Frankie here <pause dur="0.2"/> is going to crop up again he's going to we're going to go through a few <unclear> and #</unclear> <pause dur="0.2"/> and whoo in it comes <pause dur="0.2"/> just to surprise us in a little bit <pause dur="0.3"/> so it's important to know where we're going <pause dur="0.2"/> S-<pause dur="0.2"/>squared plus two-alpha-S <pause dur="0.3"/> and then alpha-squared plus beta-squared so <pause dur="0.2"/> multiplying the bit alpha-plus-J-beta alpha-minus-J-beta <pause dur="0.3"/> we're just getting rid of that # complex conjugate thing <pause dur="0.3"/> or lo and behold we can even look at it in one yet one other way <pause dur="0.8"/> which is <pause dur="0.2"/> this is in fact also S-plus-alpha-<pause dur="0.3"/>all-squared <pause dur="0.7"/> plus beta-<pause dur="0.4"/>all-squared <pause dur="0.7"/><vocal desc="clears throat" iterated="n"/><pause dur="4.0"/> A-K-A <pause dur="0.2"/> this form <pause dur="2.4"/> and

now those astute members of you that are wide awake unlike myself <pause dur="0.2"/> will note that this S-plus-alpha-<pause dur="0.2"/>all-squared <pause dur="0.3"/> is exactly what we had <pause dur="0.2"/> for <pause dur="0.4"/> the previous case <pause dur="0.2"/> the underdamped case we had <pause dur="0.2"/> didn't we <pause dur="0.2"/> didn't we have something like that <pause dur="0.2"/> the critically damped <pause dur="0.2"/> critically damped case <pause dur="0.3"/> for the critically damped case <pause dur="0.2"/> we had two roots <pause dur="0.3"/> exactly equal <pause dur="0.4"/> we did somebody's somebody's nodding yeah we had something like that <pause dur="0.4"/> well it might not have been alpha it might have been A or something but <pause dur="0.3"/> but we did did we have alpha <pause dur="0.6"/> we did <gap reason="inaudible" extent="1 sec"/><pause dur="0.3"/> the lectures actually tie together loosely <pause dur="0.5"/> so <pause dur="0.2"/> S-plus-alpha-all-squared was what we had before <pause dur="0.2"/> and now we've got this extra term the beta <pause dur="0.2"/> that's the only difference really <pause dur="0.2"/> is this extra bit which <pause dur="0.3"/> when we're factorizing it above <pause dur="0.4"/> # we can only do it in terms of the complex conjugates <pause dur="0.2"/> so that's where we are <pause dur="0.6"/> apply a unit step input <pause dur="0.3"/> this <trunc>u</trunc> here's U <pause dur="0.3"/> unit i've <pause dur="0.2"/> just to remind you it's just unit single value <pause dur="0.2"/> if it was <pause dur="0.3"/> five units then we'd just multiply the whole thing by five <pause dur="0.5"/>

just to see what happens <pause dur="2.5"/> and the output then <pause dur="0.3"/> is just the multiple of G <pause dur="0.3"/> and U <pause dur="0.4"/> or G operating on U <pause dur="2.9"/> so we've got Y here's <pause dur="0.2"/> here's G <pause dur="0.2"/> K over <pause dur="0.4"/> S-plus-alpha-all-squared plus beta-squared <pause dur="0.8"/> and then multiplied by one-over-S the step <pause dur="0.6"/> input thing <pause dur="2.0"/> business <pause dur="2.8"/><vocal desc="clears throat" iterated="n"/><pause dur="4.1"/> and now <pause dur="0.8"/> this is the bit <vocal desc="clears throat" iterated="n"/> <pause dur="0.2"/> when you close your eyes and just write and hope for the best <pause dur="0.5"/> and then we'll get back to the <pause dur="0.3"/> the easier stuff <pause dur="0.6"/> you know what we did before <pause dur="0.4"/> with partial fractions and expanded out <pause dur="0.3"/> and so on and so forth <pause dur="0.9"/> you ready for this <pause dur="0.8"/> say yes <gap reason="name" extent="1 word"/> we're ready for it </u><pause dur="1.0"/> <u who="sm0746" trans="pause"> yes <gap reason="name" extent="1 word"/> we're ready for it </u> <u who="nm0724" trans="latching"> thank you very much <pause dur="0.4"/> here we go <pause dur="0.3"/> there we go <pause dur="0.4"/> i've <pause dur="0.4"/> i spent <pause dur="0.3"/> all morning working this one out <pause dur="0.2"/> before i came in </u><pause dur="0.5"/> <u who="sm0747" trans="pause"> <gap reason="inaudible" extent="1 sec"/></u><u who="nm0724" trans="latching"> so that's right this is this is the one earlier <pause dur="0.5"/> and # you just need a <pause dur="0.2"/> squeezy bottle and three matchsticks <pause dur="0.5"/> and it all works out <pause dur="0.7"/> so the output <pause dur="0.3"/> is <pause dur="0.7"/> he said describing the difficult equation <pause dur="0.4"/> the output is K <pause dur="0.3"/> over alpha-squared-plus-beta-squared <pause dur="0.2"/> now all of these K alpha beta <pause dur="0.3"/> are just numbers <pause dur="0.8"/> a <pause dur="0.3"/> typical number <pause dur="0.4"/> is three <pause dur="1.3"/> so <pause dur="0.3"/> they might all be three <pause dur="0.3"/> in fact <pause dur="1.8"/>

and in fact if K was three she might feel a bit happier than if she was fifty-eight <pause dur="2.3"/> and then inside <pause dur="0.3"/> the <pause dur="0.2"/> parentheses <pause dur="0.7"/> or brackets or whatever you like to call them <pause dur="0.4"/> we have <pause dur="0.2"/> one-upon-S <pause dur="0.6"/> which in <trunc>fac</trunc> as # <pause dur="0.2"/> you'll note is something to do with the steady state value <pause dur="0.2"/> and it turned up up there for the unit step input <pause dur="0.2"/> and then we have two terms which look absolutely awful <pause dur="0.4"/> the <trunc>fir</trunc> both of them <pause dur="0.2"/> have the same <pause dur="0.3"/> denominator the same S-plus-alpha-all-squared-plus-beta-squared so that <pause dur="0.3"/> that's the denominator on both of those <pause dur="0.5"/> the first one is S-plus-alpha <pause dur="0.6"/> and the second one is alpha <pause dur="1.1"/> why the hell have i done it that way <pause dur="0.4"/> because i've sort of cheated a little bit and looked ahead a step <pause dur="0.4"/> # what i'm trying to do with this <pause dur="0.3"/> is to put them in <pause dur="0.3"/> partial fractions <pause dur="0.2"/> into a form <pause dur="0.3"/> that you can just go to a book and look up how they are <pause dur="0.3"/> and it tells you what <pause dur="0.2"/> that actually means in terms of time <pause dur="0.5"/> signal representations <pause dur="0.4"/> so really you can go and look up what sort of forms you can get to <pause dur="0.3"/> and then try and

chop <pause dur="0.5"/> chop this thing up <pause dur="0.4"/> # into a form that <pause dur="0.3"/> actually gives you something that you can just look up <pause dur="0.2"/> so you don't have to do any fancy stuff <pause dur="0.6"/> # <pause dur="0.3"/> don't worry when it comes to <pause dur="0.2"/> examsville <pause dur="0.4"/> this <pause dur="0.2"/> i'm not going to ask you <pause dur="0.2"/> shove this into partial fractions and look it up in a book and tell me what <pause dur="0.2"/> the answer is <pause dur="0.2"/> 'cause i'm assuming you'll all be able to do that sort of stuff if you really wanted to <pause dur="0.5"/> eventually <pause dur="0.4"/> # <pause dur="0.2"/> so that is not going to crop up <pause dur="0.2"/> so here it is but don't worry don't think blimey how the hell am i going to learn that stuff <pause dur="0.5"/> # don't <pause dur="0.2"/> panic <pause dur="0.2"/> or anything <pause dur="0.2"/> it's not going to <pause dur="0.4"/> you're not going to be asked that sort of thing <pause dur="0.3"/> you're going to be asked lots more difficult questions than that <pause dur="0.3"/> but not this particular thing <pause dur="0.7"/> what does this which is more important <pause dur="0.2"/> what does this actually look like <pause dur="0.4"/> in terms of a time signal <pause dur="4.6"/> which is the important thing <pause dur="1.7"/> time is of the essence <pause dur="1.5"/> what the hell does that mean <pause dur="1.2"/> i thought essence was sort of <pause dur="1.6"/> some spice or something like that <pause dur="0.5"/>

what's time got to do with being spicy </u><pause dur="1.6"/> <u who="sm0748" trans="pause"> <gap reason="inaudible" extent="1 sec"/> </u><pause dur="0.4"/> <u who="nm0724" trans="pause"> no <pause dur="0.4"/> perhaps it's something to do with Geri Halliwell i don't know <pause dur="1.7"/> right <pause dur="1.7"/> or maybe that's just wishful thinking again <pause dur="1.1"/> this is just to confuse # all of you never mind those listening in <pause dur="0.3"/> here's the gain so this is the output <pause dur="0.3"/> in terms of time <pause dur="0.9"/> this is what it what it actually looks like what the <trunc>syste</trunc> for a <pause dur="0.3"/> an underdamped system <pause dur="0.4"/> what it would actually look like the response that you get <pause dur="0.9"/> would would appear like <pause dur="0.4"/> if it is <pause dur="0.3"/> the Citroen Diane that i spoke of earlier <pause dur="0.3"/> it would be the position of the shock absorbers after a <pause dur="0.4"/> a bump or something like that after a hump <pause dur="1.2"/> or hump's not a good word to use <pause dur="1.3"/><vocal desc="laughter" iterated="y" dur="1"/><pause dur="0.5"/> not in a Citroen Diane anyway <pause dur="1.7"/><vocal desc="laughter" n="ss" iterated="y" dur="1"/> i don't know might be possible <pause dur="4.5"/> this <pause dur="0.2"/> probably if this <pause dur="0.4"/> tape gets sent to the Vice Chancellor this will be the last lecture <pause dur="0.3"/> that i'll be giving <vocal desc="laughter" iterated="y" dur="1"/><pause dur="3.9"/> there'll be <pause dur="0.4"/> women's rights campaigners <pause dur="0.4"/> men's rights campaigners Skoda rights <vocal desc="laughter" iterated="y" n="ss" dur="1"/> campaigners there'll be all sorts <vocal desc="laughter" iterated="y" dur="2"/><pause dur="0.3"/> # <pause dur="1.0"/> the <trunc>anti</trunc> <pause dur="0.2"/> anti-Slovak lobby or something probably the <pause dur="0.2"/>

probably some refugees will crop up somewhere <vocal desc="laughter" iterated="y" n="ss" dur="1"/> in there as well <pause dur="0.7"/> # <pause dur="0.3"/> K all complaining <pause dur="0.3"/><shift feature="voice" new="laugh"/> K over <shift feature="voice" new="normal"/> alpha-squared-plus-B and then there'll be people complaining that they weren't included <pause dur="0.3"/> K over alpha-squared-plus-beta-squared is the same gain that's just come through <pause dur="0.2"/> the whole thing just like a dose of prunes there it is again <vocal desc="laughter" n="ss" iterated="y" dur="1"/> nothing's affected it whatsoever <pause dur="0.5"/> # <pause dur="0.2"/> and then inside the bracket here's that one which was the one-over-S there it is again <pause dur="0.2"/> and two terms <pause dur="0.4"/> # <pause dur="0.5"/> exponential-<pause dur="0.2"/>minus-alpha-T <pause dur="0.2"/> multiplied by the cosine of beta-T <pause dur="0.6"/> and the same sort of thing again exponential-minus-alpha-T multiplied by the sine of beta-T <pause dur="0.7"/> and all we got is that that's just <trunc>m</trunc> little gain A over alpha-over-beta <pause dur="0.5"/> which are just numbers remember alpha beta and K are just numbers <pause dur="0.6"/> and the nice little thing here you'll see the alpha and the beta <pause dur="0.5"/> i find these little things nice <pause dur="0.4"/> the exponential has the alpha bit in it <pause dur="0.3"/> so there's no beta in the exponential <pause dur="0.4"/> and the beta turns up <pause dur="0.4"/> with the sines and

cosines the frequency type of the the changing element <pause dur="0.3"/> and there's no alphas <pause dur="0.2"/> knocking around with the betas <pause dur="0.4"/> so the alphas and the betas are sort of # separated <pause dur="0.4"/> # and they they turned up in <pause dur="0.2"/> a different form in the <pause dur="0.4"/> the denominator <pause dur="0.3"/> and here they're having different effects in the output <pause dur="0.3"/> but <pause dur="0.4"/> can we can we see that in another way <pause dur="0.2"/> well we can this this will <pause dur="0.5"/> hopefully make it a little bit clearer <pause dur="0.4"/> the only thing <pause dur="0.4"/> i mean it may be first instance look a little bit more complicated but in fact it simplifies <pause dur="0.6"/> when we're looking at what happened <pause dur="0.4"/> this can be written by <pause dur="0.4"/> playing around with <trunc>so</trunc> sine and cosine <pause dur="0.8"/> what do they call trig identities or something like that is it <pause dur="1.2"/><vocal desc="clears throat" iterated="n"/><pause dur="1.3"/> sounds like something out of Only Fools and Horses whatever <pause dur="2.5"/> and we've got the gain that's still there it's sort of lingering in there it is <pause dur="0.4"/> K over alpha-squared-plus-beta-squared no change there <pause dur="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 <pause dur="0.5"/> # <pause dur="0.2"/> i've cheated a little bit because <pause dur="0.4"/> there should be some sort of phase <pause dur="0.3"/> shift <pause dur="0.3"/> that appears <pause dur="0.5"/> in the <trunc>ins</trunc> it's not really just beta <pause dur="0.3"/> there should be a phase shift <pause dur="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 <pause dur="0.4"/> and you're you're looking at this think blimey this should be a phase shift he's got it all wrong <pause dur="0.2"/> well <pause dur="0.4"/> i i'm just excluded it <pause dur="0.3"/> or <trunc>execu</trunc> got rid of it just for the purposes of going through this <pause dur="0.3"/> there is a bit of a phase shift in there <pause dur="0.3"/> but it's not really important in terms of <pause dur="0.3"/> what how we're looking at this <pause dur="0.6"/> # later on <pause dur="0.4"/><vocal desc="clears throat" iterated="n"/> it may become important but not at this stage <pause dur="1.5"/> so we've got the gain <pause dur="0.4"/> # that that one is still there which multiplies by the gain and then we've got this time dependent thing <pause dur="0.4"/> that in here is <pause dur="0.2"/> the square root <pause dur="0.4"/> of <pause dur="0.2"/> alpha-squared-plus-beta-squared <pause dur="0.3"/> on the top <pause dur="0.4"/> divided by beta <pause dur="0.6"/> that's that which again is just a gain component just a number <pause dur="0.3"/>

and then we've got the two terms <pause dur="0.3"/> exponential-minus-alpha-T <pause dur="0.3"/> multiplied by sine-beta-T <pause dur="0.9"/> and again the exponential has the alpha <pause dur="0.4"/> and the sine has the beta <pause dur="0.3"/> and the two things are separate <pause dur="0.4"/> so <pause dur="0.4"/> what does this all look like <pause dur="0.3"/> # well couldn't be bothered to work it out myself <pause dur="0.3"/> so i thought <pause dur="0.2"/> let you do a bit of work i thought make this an interactive lecture <pause dur="2.4"/> the steady state value <pause dur="0.5"/> # this is as time goes off to infinity <pause dur="0.7"/> again <pause dur="0.2"/> when time is B <pause dur="3.6"/> well what is it <pause dur="0.3"/> can we work it out <pause dur="0.2"/> can we see </u><pause dur="4.6"/> <u who="sm0749" trans="pause"> <gap reason="inaudible" extent="2 secs"/></u> <pause dur="0.3"/> <u who="nm0724" trans="pause"> oh you've given it away <pause dur="3.1"/> yeah <pause dur="0.3"/> if <pause dur="0.2"/> if you set <pause dur="0.7"/> T to be zero <pause dur="0.4"/> in that lot <pause dur="0.7"/> then <pause dur="0.3"/> the <trunc>overa</trunc> it doesn't really matter what the sine is doing that's going up and down as sines do <pause dur="0.5"/> but this this thing exponential of <pause dur="0.4"/> minus-infinity will be zero <pause dur="0.4"/> and that wipes the floor with anything else <pause dur="0.4"/> so # as time goes off to infinity we've got this gain <pause dur="0.3"/> multiplied by one-minus-zero <pause dur="0.4"/> and one-minus-zero is </u><pause dur="1.2"/> <u who="sm0750" trans="pause"> one </u><pause dur="0.5"/> <u who="nm0724" trans="pause"> one who said zero <pause dur="0.7"/> one <pause dur="0.2"/> yeah <pause dur="1.0"/><kinesic desc="writes on transparency" iterated="y" dur="6"/> so it's just K <pause dur="0.5"/> over alpha-squared-plus-beta-squared is the steady state value <pause dur="2.2"/> in fact <pause dur="3.3"/><kinesic desc="changes transparency" iterated="y" dur="6"/> in fact if we

went <pause dur="0.6"/> back <pause dur="0.8"/> just out of interest <pause dur="0.2"/> if we went back here <pause dur="0.5"/> and had a look <pause dur="0.4"/> at at this thing <pause dur="0.3"/> if # we remember with the step input <pause dur="0.2"/><vocal desc="sneeze" n="su0751" iterated="n"/> if we just bless you <pause dur="0.2"/> if we just set S to zero <pause dur="0.2"/> and multiply this thing out <pause dur="0.4"/> then which is <pause dur="0.2"/> alpha-squared-plus-beta-squared K over alpha-squared-plus-beta-squared <pause dur="0.4"/> so we we can just look at the thing in the first place <pause dur="0.3"/> and if you remember the steady state value for a step input <pause dur="0.3"/> we just set S to zero <pause dur="0.3"/> and we've got our answer and that's the same doing the same thing there <pause dur="0.5"/><kinesic desc="changes transparency" iterated="y" dur="18"/>

and that's what we've got <pause dur="0.3"/> K over alpha-squared-plus-beta-squared <pause dur="1.4"/> but let us this this denominator <pause dur="0.2"/> i threw in <pause dur="0.2"/> # a few minutes ago <pause dur="0.3"/> just to <pause dur="0.4"/> to add a bit of <pause dur="0.2"/> spice and <pause dur="0.2"/> <trunc>ele</trunc> <trunc>some</trunc> something there 'cause it's it's something <pause dur="0.3"/> we can sort of ask exam questions on sometimes <pause dur="0.5"/> and i want to look at it a little bit more closely <pause dur="1.7"/> so let us <pause dur="0.8"/> let us return to the denominator <pause dur="3.5"/> the denominator is the bit on the bottom <pause dur="2.6"/> you should know <pause dur="5.8"/> the numerator's the bit on the top <pause dur="1.6"/> and in the middle <shift feature="voice" new="laugh"/> well <shift feature="voice" new="normal"/><pause dur="0.2"/> keep your eyes closed <pause dur="1.4"/> you've got to draw the line somewhere there <pause dur="1.3"/> ha ha ha ha <vocal desc="laughter" iterated="y" dur="1"/> <pause dur="0.7"/> so the numerator <pause dur="2.4"/> is commonly <trunc>thi</trunc> <pause dur="0.2"/> this is the bit on the right hand side <pause dur="0.4"/> see i <trunc>writ</trunc> i wrote the right hand side first just <pause dur="0.4"/> so you what the hell's he doing <pause dur="0.5"/> and on the right hand side this is what we had before <pause dur="0.3"/> S-squared plus two-alpha-S <pause dur="0.3"/> plus <pause dur="0.2"/> alpha-squared-plus-beta-squared that's what we had before <pause dur="0.2"/> for the denominator <pause dur="0.8"/> # <pause dur="0.3"/>

well about <pause dur="0.4"/> ten lines ago <pause dur="0.3"/> was it not <pause dur="0.3"/> was it not <pause dur="2.4"/> and <pause dur="0.5"/> this is just to say that <pause dur="0.8"/> this this left hand side is how it is <pause dur="0.3"/> often written <pause dur="1.3"/> and we'll see why in a second <pause dur="0.5"/> so the S-squared <pause dur="0.4"/> has not changed it's S-squared <pause dur="0.5"/> and then we've got plus two-<pause dur="0.7"/>zeta-<pause dur="0.8"/>omega-<pause dur="0.3"/>baby-N or subscript-N-<pause dur="0.4"/>S <pause dur="0.6"/> plus omega-N-squared <pause dur="2.5"/> and again <pause dur="0.2"/> zeta <pause dur="0.2"/> we're trying to bring in <pause dur="0.2"/> people <pause dur="0.3"/> half the names that are nearly married to American film stars things like that <pause dur="0.5"/> so it sort of adds a bit of artistic licence to what we're doing <pause dur="0.6"/> so two-zeta-omega-N-S <pause dur="0.6"/> plus <pause dur="0.3"/> omega-N-squared <pause dur="0.6"/> and we can see that i mean straight even i <pause dur="0.3"/> could do this <pause dur="0.2"/> this morning <pause dur="0.3"/> with before my coffee <pause dur="0.6"/> zeta is <pause dur="0.6"/> if we just look at this we equate <pause dur="0.5"/> S-squared equals S-squared <pause dur="1.2"/> there's a profound statement <pause dur="0.5"/> and then if we equate the S terms <pause dur="0.4"/> two-zeta-omega-N <pause dur="0.5"/> is two-alpha <pause dur="1.2"/> and the twos <pause dur="0.4"/> cancel <pause dur="2.0"/> which leaves us zeta is <pause dur="1.0"/> alpha-over-omega-N <pause dur="0.2"/> if we want <pause dur="0.4"/> and also the omega-N-squared is alpha-squared-plus-beta-squared <pause dur="1.3"/> that's just so <pause dur="0.2"/> so what have we got this is where we came from

the right hand side <pause dur="0.3"/> and all i'm saying here is well <pause dur="0.4"/> you know if you go out in the big world the wide boys down Oxford Street <pause dur="0.2"/> they wouldn't talk in terms of alphas and betas <pause dur="0.3"/> they would talk in terms of zetas and omega-Ns <pause dur="1.0"/> well they maybe wouldn't <pause dur="0.2"/> even talk like that <pause dur="0.2"/> but that that's the the common terminology is that one <pause dur="0.5"/> and as we can see <pause dur="0.2"/> let us let us look <pause dur="0.3"/> at extremes let us look at extreme cases <pause dur="0.4"/> what for instance <pause dur="0.3"/> happens <pause dur="0.2"/> when beta <pause dur="0.6"/> tends to zero <pause dur="1.3"/> what goes down what happens <pause dur="2.6"/> and it's # <pause dur="0.3"/> it's intriguing really isn't it when beta goes to zero <pause dur="0.3"/> 'cause as beta goes to zero <pause dur="0.7"/> if you're looking at at this side here <pause dur="0.7"/> we're we're back to the critically damped <pause dur="0.3"/> case <pause dur="0.4"/> when beta heads off as beta gets to zero we're back to the critically damped case <pause dur="0.5"/> # the roots are just <pause dur="0.5"/> S-plus-alpha-<pause dur="0.6"/>squared <pause dur="0.5"/> all-squared <pause dur="2.4"/> just like what <pause dur="0.3"/> they were <pause dur="0.2"/> sort of thing before <pause dur="13.0"/> and and now for the <pause dur="0.2"/> the foreign listeners sorry <pause dur="0.2"/> <distinct lang="fr">maintenant</distinct> <pause dur="0.4"/> # <distinct lang="fr"> nous</distinct> <pause dur="0.3"/> will set zeta <pause dur="0.6"/> to <distinct lang="fr">un</distinct> <pause dur="0.5"/><vocal desc="laughter" n="sl" iterated="y" dur="4"/> <pause dur="2.8"/> and there it is <pause dur="1.1"/> <shift feature="voice" new="mimicking French accent"/><distinct lang="fr">le </distinct> damping ratio <shift feature="voice" new="normal"/><pause dur="1.0"/> <vocal desc="laughter" iterated="y" dur="1"/> <pause dur="0.3"/> so zeta

zeta is actually known <pause dur="0.8"/> A-K-A <pause dur="0.3"/> the damping ratio <pause dur="0.9"/> and is as we set it to one <pause dur="1.1"/> it's it's all the same thing here <pause dur="0.2"/> if you look here <pause dur="1.4"/> it's all magic <pause dur="0.3"/> beta goes to zero <pause dur="0.2"/> so omega <pause dur="0.3"/> tends to alpha <pause dur="0.6"/> and as omega tends to alpha so zeta <pause dur="0.2"/> alpha-over-alpha <pause dur="0.6"/> is one <pause dur="1.2"/> here so <pause dur="0.2"/> just just from these equations <pause dur="0.2"/> beta to zero <pause dur="0.6"/> omega <pause dur="0.2"/> becomes alpha <pause dur="0.2"/> effectively <pause dur="0.6"/> alpha-over-alpha <pause dur="0.4"/> is just is one <pause dur="0.3"/> so zeta tends to one <pause dur="0.3"/> so this thing <pause dur="0.2"/> the damped zeta is known as a damping ratio <pause dur="0.3"/> and as beta goes to zero <pause dur="0.4"/> zeta the damping ratio goes to <pause dur="0.3"/> to a value of one <pause dur="1.4"/> and <pause dur="0.2"/> the other and and really what beta <pause dur="0.5"/> was linked <pause dur="0.2"/> to the sine term <pause dur="0.3"/> the sine term going # really the oscillations <pause dur="0.4"/> so what this is saying effectively <pause dur="0.3"/> is as the oscillations die down <pause dur="0.2"/> as beta goes to zero the frequency of oscillations dies down and so on <pause dur="0.4"/> the damping ratio tends to one <pause dur="0.3"/> and we tend to the critically damped case where we don't have oscillation <pause dur="0.5"/> so it all sort of ties together really <pause dur="1.1"/> the # another extreme as alpha <pause dur="0.4"/> tends to zero <pause dur="9.5"/> then <pause dur="1.3"/> we can see see from here as

alpha's zero then omega-N <pause dur="0.5"/> becomes beta <pause dur="0.3"/> or omega-N-squared is beta-squared <pause dur="2.0"/> and this omega-N <pause dur="0.3"/> which is why we've got this subscript-N <pause dur="0.8"/> is also known as the natural frequency <pause dur="8.2"/> and for this <pause dur="0.2"/> for this reason effectively <pause dur="0.5"/> # <pause dur="2.3"/> as beta goes to zero <pause dur="0.4"/> # as alpha goes to zero omega-N <pause dur="0.2"/> tends to <pause dur="0.2"/> to beta <pause dur="0.3"/> becomes <pause dur="0.2"/> beta <pause dur="0.5"/> and <pause dur="0.3"/> # with alpha being zero it means zeta <pause dur="0.5"/> is zero <pause dur="0.9"/> so and zeta being zero we've got no damping <pause dur="0.3"/> we've we've gone in the direction of the <pause dur="0.3"/> pure <pause dur="0.3"/> oscillations <pause dur="1.3"/> we we switch the thing on <pause dur="0.3"/> and it just oscillates it's a pure oscillator <pause dur="2.2"/> it just goes up and down up and down up and down <pause dur="0.2"/> never gets exhausted never gets tired <pause dur="2.0"/> never dies down <pause dur="0.2"/> just keeps going <pause dur="3.3"/> it's a bit like Bolton Wanderers <pause dur="0.2"/> really <pause dur="0.2"/> just keep going up and down <pause dur="0.4"/> never <pause dur="1.0"/> never settle down <pause dur="1.1"/> and # </u><u who="sm0752" trans="overlap"> <gap reason="inaudible" extent="1 sec"/></u><pause dur="0.6"/> <u who="nm0724" trans="pause"> pardon <pause dur="1.8"/> oh <pause dur="0.2"/> <vocal desc="laugh" iterated="n"/> <pause dur="0.3"/> you're not a Bolton fan are you <pause dur="1.3"/> sorry about that i didn't <pause dur="1.0"/><vocal desc="clears throat" iterated="n"/> as opposed to <gap reason="name" extent="1 word"/> which is just going down and down so <pause dur="3.0"/> and so for the underdamped case this is the whole <sic corr="caboodle">skaboodle</sic> <pause dur="0.7"/> this term zeta <pause dur="0.8"/> will <pause dur="0.4"/> go from zero <pause dur="1.0"/> in the case

when there's no damping we've just got pure oscillations <pause dur="0.4"/> to one <pause dur="0.3"/> in the case where we were at before <pause dur="0.2"/> the critically damped case <pause dur="0.7"/> and <pause dur="0.2"/> realistically as the term zeta <pause dur="0.4"/> tends towards zero so the oscillations get bigger there's more of them <pause dur="0.4"/> as zeta gets <trunc>w</trunc> towards one <pause dur="0.5"/> then the oscillations get smaller <pause dur="0.3"/> so so <pause dur="0.2"/> zeta is just a <pause dur="0.3"/> a knob that you can twiddle <pause dur="0.2"/> effectively <pause dur="0.3"/> and between zero and one <pause dur="0.3"/> and as you twiddle it <pause dur="0.4"/> so oscillations get bigger or smaller <pause dur="1.3"/> just just to conclude this and then we'll have a <pause dur="0.5"/> # a breather <pause dur="2.3"/> it's really bringing things together the extremes <pause dur="0.2"/> when zeta is one <pause dur="0.2"/> we've got the case of critical damping <pause dur="0.2"/> that's <pause dur="0.5"/> i know you all know that but i thought i'd write it down just for the sake of completeness <pause dur="0.7"/> and in fact <pause dur="0.3"/> when zeta <pause dur="0.3"/> is greater than one <pause dur="0.6"/> it's not really

what we're looking at but when it's greater than one <pause dur="0.3"/> we're off back to the overdamped case <pause dur="14.1"/> and <pause dur="0.7"/> again the extremes the other point was when <pause dur="0.4"/> zeta is zero <pause dur="1.1"/> that's these <pause dur="0.2"/> there's natural frequency <pause dur="0.4"/> oscillations omega-N <pause dur="0.3"/> and they just go on forever and ever and ever and <pause dur="0.7"/> don't die down <pause dur="12.4"/> and i think at this juncture <pause dur="0.9"/> how's about we have <pause dur="0.2"/> 'cause # this is a double header this morning for me isn't it <pause dur="0.5"/> i think </u><pause dur="0.4"/> <u who="sm0753" trans="pause"> yeah </u><pause dur="0.3"/> <u who="nm0724" trans="pause"> hopefully yeah <pause dur="0.4"/> i'm glad you're giving me support in that yeah <pause dur="0.6"/> so # <pause dur="0.3"/> maybe # as i fancy making myself a cup of coffee <pause dur="0.3"/> if we said five past <pause dur="0.7"/> is that all right five past and we'll get going <pause dur="0.4"/> and we'll get this wrapped up <pause dur="0.3"/> so i'll see you all then have a wonderful time <pause dur="0.3"/> in the fifteen minute break <pause dur="0.5"/> or fourteen minutes </u></body> </text></TEI.2>