Demo a Course Session

Duration:"00:50:39"

00:00:15:15 OK, how are we doing? 

00:00:19:11 How are you? 

00:00:23:09 Maybe the snow's going away. That's a good thing. One can hope. Could be worse. We could be back East. I hear it's really bad. 

00:00:33:11 Ready? We have virtual attendance. See, that's not bad. 

00:00:36:15 And then it's ice. And I just-- I talked to a-- was supposed to meet with the guy for lunch today. And he's stuck in Columbus, Ohio. He's not even sure if his flight's going to get-- plane's going to get out today. So, OK, questions? 

00:00:56:21 Alex? 

00:00:57:01 I'm going to say this probably poorly, but is there something called a indemnification clause about being-- from contract-- your contract-- I know I'm saying it wrong. 

00:01:09:17 No, indemnification, so in general, it's whoever is indemnified, they're protected against legal actions under certain circumstances. So it's like indemnified and hold harmless. So that if a client-- I mean, we're-- I'm trying to give a decent example. 

00:01:40:25 If you're in a contractual relationship but the other party behaves in a certain way that's reckless and irresponsible, there would be, presumably, clauses in the contract that would indemnify you from harm if things were brought forth. So I mean, just things like contractors are responsible for putting together storm water, erosion control plans. We would review those. But if it wasn't properly maintained, we'd be indemnified, I guess. 

00:02:21:16 Was this a particular-- just something that popped into your mind? I was reading contract docs this morning. And part of the construction fitting, construction-- I think contract docs that said we'd do some observation had that clause in there at the end. So that way, I think if there's something wrong, we, as engineers, wouldn't get sued. The city would pay for our indemnification of that. I was trying to figure out what that-- 

00:02:45:05 So it would just be protecting and holding harmless for actions by others that we wouldn't be held responsible for, outcomes that happened. And it may come up again as we-- as we get more into the general conditions. Let's make sure we try and maybe catch that. OK, good. Other questions. 

00:03:15:24 So I was going to share with you the Lewiston project here. We'll just go to overhead. I'll show a few things. As I've already told you, it's Lewiston, so that-- so here's the bid form, just to kind of get a sense of it. You can see in this case, your opening bids. There were four addenda. so when it was received, the number, initials, those types of things. 

00:03:41:17 But what went South in this project is this particular language here. So this is under vertical turbine pumps. So a water pump, this was a water booster pump station. Under work included, so this is a technical spec, we haven't arrived at technical specs yet, but we will. And so they say in here, work included, "as a bid alternate, the contractor may propose vertical inline pumps." 

00:04:12:04 And then the spec points the bidder to a particular section for the 50 and 100 gallon minute application only. If selected, the inline pumps shall be installed over the can. So these were pumps, cans that were already in place at the pump station, which should be covered with a blind flange as shown in the drawings. 

00:04:33:15 The base-- so then it does say the base bid shall be for providing vertical turbine can pumps, which was the original spec one. And then any costs associated with providing and installing inline pumps shall be borne. So you read that, so you say, first you say, as a bid alternate, and then-- but then it clarifies then it says, well, but the base bid is for the vertical turbine can pumps, not the vertical turbine inline pumps, so slightly different. 

00:05:08:01 So does allow for that. Not language I would have chosen. It led to, let's see, so here's the bid. So it was just it-- it was a lump sum project. Yeah, so the bidder shall list below the total lump sum bid prices. So this was a $1.8 million lump sum project, which is beyond my comfort range. That's a lot of money for a lump sum. 

00:05:42:28 But no opportunity. It doesn't speak to base bid. It doesn't speak to alternate. But the technical specifications, which are part of the bidding documents, did speak to that. So they wrote in there their dollar amount. Then they said deduct $98,700 if vertical inline pumps are used. So they just wrote it in there because it was referred to in the technical specification. 

00:06:07:18 So the bid form didn't account for a base bid. Didn't list alternates. And we've looked at the bid form in your document. And it's got that type of structure. So contractor, a very creative smart contractor, saw that opportunity. If we look at the total prices here, I'm not going to give you the names, although it really doesn't matter. This was several years ago. 

00:06:33:19 And so you can see, so there's the-- there was their bid. It took them down to $1.719 million. You can see two other contractors observed the same, but they were not-- it wouldn't have dropped them to low bid. But you can see, so this is low bid, the apparent low bid with the deduct. Contractor 2 didn't deduct. And so you can see $1.818 versus $1.750. So this is number one, versus number two. Number one didn't list that. Number two did, these ones. 

00:07:07:00 So now they're in a pickle, right? Because the technical specifications stipulated base bids, stipulated alternate. The bid form didn't include for it. Contractors address that, some of them. I won't-- I'll just read you from the letter. So during the bid process, so this is a letter from the contractor to the city, because they're hearing the bids might be rejected. 

00:07:34:27 During the bid process, we conducted a thorough analysis of the bid contract documents provided by the city and project engineer of record, which again, is required in the instructions to bidders, in the bid form, thorough analysis. In the case of this project, the contract specification stated a bid alternative at the contractor's choice. It was clear at the contractor's choice. 

00:07:55:00 This is clearly indicated in spec section, as copied below. And then the specs acknowledge the alternate is that the contractor's choice and not a requirement for the bid proposal. They found it in the best interest of the city. So they tried to make the compelling argument, you opened the door, you stipulated it, and then-- and then you're in this-- now you got yourself in a bind, because the bid form doesn't account for it. 

00:08:21:07 And they didn't stipulate in the instructions to bidders how they might address that. So all of those things should have been covered. This is construction bidding 101 quite frankly. And the outcome was a rejection of all bids on basically a $2 million project. My understanding is the next time around, prices went up. And so it costs the city. And that low bidder, I mean, there are several good contractors out there. 

00:08:48:26 That one in particular is a very good contractor, didn't bid. Walked away from it. And so there are consequences to our failures in putting together bid documents. And so, again, that's just 101. If you're going to develop a tech spec that allows for an alternative, you need to immediately put a note in your bid form somehow to catch it later on as you're developing that. 

00:09:14:04 So that was a pretty bad one. And it cost-- it cost the city money. It cost the engineer money. So anyway, interesting one. So I'm sure there's more out there like that, unfortunately. So questions? Yeah. 

00:09:32:11 What do you think is the leading factor for having-- for spec? Do you think is the transportation costs, like hauling equipment, or labor shortage? 

00:09:45:29 So let me-- so we better understand your question, and I'll try and repeat for EOC. What actually in-- 

00:09:51:16 When bidding out a job, what is the biggest factor that makes it cheapest compared to more expensive? 

00:09:58:23 Like for a contractor? I mean, it's going to vary by type of project, street project, underground utility project, where you're located. Mobilization costs are obviously expensive. Pipe materials can be expensive. Excavation is pretty fixed, probably. It's just equipment and labor. Pipe materials can vary substantially. 

00:10:22:10 In today's world, I mean, who knows? Asbestos-- or asphalt concrete, any of the materials for road construction. So I'd say it's highly-- it's going to vary project to project on what drives the costs. And proximity-- certainly mobilization costs can affect things. Although we, again, we typically cap that as-- and most contractors will come in at, say, 5% pretty close to the cap on mobilization, et cetera, demobilization. 

00:10:56:14 There's no one thing, though. So I'm not I'm not waffling. It's just highly variable, depending on the project. And these-- but these are the things that we do our best to stay on top of. It's one of the many hats we wear as engineers. Is we're doing the technical design, technical specifications, contract, thinking about pricing. 

00:11:16:26 Making sure we're not gold-plating projects. I mean, this situation, this project in Colfax, now we're finding out that with new energy efficiency requirements in the state of Washington, it's requiring pump suppliers to upsize motors in order to get in the motors of a similar efficiency curve. And they're not maximum efficiency at maximum horsepower. 

00:11:41:28 And so here we are, we've pulled out a system that's 75 horsepower. And unless we want to derate the capacity of the well, which really isn't desirable, we're probably going to have to bump to 100 horsepower, because you don't-- there isn't an 80 horsepower or 85 horsepower motor. It's 75 to 100. 

00:11:59:09 So now we're in this-- we hadn't seen this coming. It was-- and trying to-- first thing you do is you let the client know, hey, energy standards in Washington might bump us to 100 horsepower. But then we got to go back and we got to look at all the electrical equipment. And what sort of domino effect does that have on the design and the cost, right? 

00:12:22:29 It's a difficult-- it's an increasingly difficult environment to project design projects. OK, other questions? OK, so we left off here on this slide 29. This is unit pricing and contract management. 

00:12:50:04 And this is one, I'm going to do my best to explain or give you an example on cash allowances. I'm going to-- you don't need to bounce to 14.03, but I will really quickly. If I go to instructions to bidders, get rid of that. 14.03, there's a little-- so in the instructions to bidders, for cash allowances, the bid price shall include such amounts as the bidder deems proper for contractor's overhead, et cetera. 

00:13:23:15 So they're stipulating what's in a cash allowances piece. And then are there extra handouts? You want to pass one back to the back? Just pass one, just float one back. Oh, OK, OK, good. So it was here, and then gone, and then here? Somebody saved it for you? OK, that's good. That's good. 

00:13:50:10 You got some friends back there. That's good. Because I wasn't going to pay one for you. And you can't ask for it in the next class either. I'm kidding. So here, so it's spelling out to the contractor, if there is a unit price contract, a line item for allowances, you would spell those out in your technical specifications. 

00:14:14:24 And then it's also further covered. And we will spend some time-- I'll hit on Article 11 right now. But we'll come back to it more than once, because there is a lot of information there. But if we go to 11.02. Again, you don't need to bounce around here. Shortly, maybe we'll jump there for a while. 

00:14:34:14 But this cash allowances contractor agrees. So in there, yeah? Sure. So I jumped to general conditions article 11, 11.02. sorry, I was-- it's one of the challenges, slides, I don't want to cut a whole bunch of stuff from the EJCDC into the slides. So we have to bounce back and forth. And I feel like Logan's still hasn't tapped his. 

00:15:01:16 I honestly don't know what to do-- 

00:15:04:09 OK. 

00:15:04:19 What time is it? 

00:15:05:25 OK. 

00:15:10:04 Yeah, so here's where it's spelled out, everything. And actually, article 11 in the general conditions, we'll come back to. Because one of my former grad students shared with me a really useful story about how to-- experience on how to deal with things like this. But here, contractor agrees that cash allowances include the cost to contractor of materials, equipment required by, to be delivered to the site, and all applicable taxes, and contractor's costs for doing all these things. 

00:15:39:27 So it's right there in the general conditions on everything that needs to be the contractor if there is something for cash allowances. And so, sorry, we'll bounce back to the slides for a minute. What is it? It's a lump sum that you would put. You would just have a line item and a unit price contract. You would pay it off typically on a lump sum basis you can pay percent of, I suppose, that provides the owner with a general item to pay contractor for extras. 

00:16:12:17 Maybe there's some uncertainty during the design, the contractor-- the owner's not quite sure what they might want to purchase, so funds to buy stuff later. And it could be-- I mean, it could be instrumentation. Hey, they'd like to-- we had a-- when we were closing, I wasn't involved with it, when the wastewater treatment plant project in Culdesac was being closed out, they had some money left over. 

00:16:41:07 And so I helped the engineer work to buy some instrumentation. So they could do some of their own analytical work. They bought a spectrophotometer for nitrogen testing, which was pretty useful. They had some funds left over. So those types of things, you could envision that. Maybe you put it as an alternate in your bid. 

00:17:01:26 But it's just that-- it just gives you some, if the owner wants, if there's maybe some things that might be desired, it allows the owner to have the contractor buy stuff, whatever that stuff may be. Now, that all being said, I mean, not that I've bid out hundreds of projects, but I've never used cash allowances. 

00:17:24:17 But it's there. It's there. It's nice to know. It's good to know that you have that flexibility, that you could write that into a contract. And the general conditions cover that. So you're well covered if you want to do that. 

00:17:41:08 Let's see, go to the next one, OK unbalanced bid. So this is an interesting one. And this is the handout. We'll take a glance at that. So what is it? So this is overpricing one or more items in a unit price bid. So this wouldn't be in a lump sum, because a lump sum is one number. 

00:18:02:23 It's often called penny bidding. And we'll see an example where it literally was penny bidding. Why might a contractor do this? They might employ unbalanced to conceal some pricing strategies from contractors. Maybe there's-- they see an opportunity where they can frontload something, where they get a lot of money out of-- they sort of-- basically, what they're doing is they're putting together their whole bid, putting together their bid with all the unit price items. 

00:18:36:06 And then they might end up-- they might see up front that there's maybe some uncertainty in quantities, the engineer screwed that up. So they can put a large number by that. And then-- and then balance it out somewhere else by putting a low unit price for an item that they then can be compensated. So they'll actually make a lot of money out of one particular item. So it's unbalanced. It's not equitably bid. 

00:19:05:19 Again, believing the quantity estimates are inaccurate. So there might be a windfall. So you jack up that particular price. But in order to secure the bid or hopefully win the bid, they would elsewhere put in a really low price on an item. And there is some case law, we're not going to jump into that, other than just a case study. 

00:19:26:19 Now, one of the things is, it's obviously-- if you go look at IB 19.01, and I'm not-- I'm not going to drag you there, I'm going to go there myself, but just point something out. It's minor, but it's I've covered it in the slides. If we look at 19.01, owner reserves the right to reject any and all bids, including unbalanced. 

00:19:54:27 So we are-- so it's stipulated in the instructions to bidders if it's unbalanced, that the owner can reject based on that. Yup? 

00:20:07:26 How do they determine whether it's unbalanced? 

00:20:09:21 (LAUGHS) 

00:20:12:21 No, no, so good question, how do they determine? So not from the engineer's estimate, you just look at it. So let's just take a minute and look at this one I gave you. See if you-- just kind of-- anything look-- take a few minutes and just look at this handout here, this transmission line project. We've got five bids in there, plus the engineer's estimate. So obviously, I was not penny bidding. So you can ignore the engineer's estimate. 

00:20:44:07 But-- So you think item 16, additional cost for-- 

00:20:54:27 He estimates $200, but you put $24. 

00:20:59:06 So you're saying I was penny bidding? You think I'm unbalanced? 

00:21:02:17 No, I thought it would be-- jacking down that one. 

00:21:05:20 But remember, so this is-- so this is a good one to point out. So Rock ex, we didn't this-- I gave that example earlier. We didn't know if we would have rock ex. We had 2 and 1/2 miles of pipeline. We wanted to be able to pay the contractor if they encountered-- so we just said, gosh, we think there's 100 cubic yards of rock excavation. 

00:21:27:11 And let me further clarify, in the payment section, in the measurement and payment section, it is made very clear what is rock excavation. There is certain equipment that has to be used if the rock is to-- so if the rock can be just-- rock excavation isn't rock, isn't just all rock. It's very specific to-- it would spell out the type of equipment necessary to excavate said material. 

00:21:56:09 That would qualify under rock excavation. So we throw-- we just throw one in there. And rock ex can be expensive, because it's expensive equipment. And so remember Logan, the point is-- so here, this would be a situation where you might interpret penny bidding or unbalanced bidding, so maybe they would increase the price substantially, right? 

00:22:24:04 Because they're like, wow, we actually think that it's going to be a lot. And so we see an opportunity to make some money. And so then elsewhere, they would reduce the price. In this case, given the uncertainty, we hadn't-- we hadn't shown on the drawings. If you knew somewhere where there was rock, you would spell it out. 

00:22:42:15 You would provide that information in your supplementary general conditions. You would provide that information. You would provide geotechnical information in the bidding documents. We didn't do that. We just sort of put it out there as speculative, just to cover it. So in this case, it wasn't deemed unbalanced. And we didn't see anything that would suggest penny bidding elsewhere to-- because the low contractor was Skyline there. 

00:23:11:25 They were at $20. That's, wow, that's cheap. You want to rock ex for us for $20, we're OK with that. That's a good price. It's on the contractor. It's not our fault that you underbid or underpriced that particular item. Anything else sort of jump out? 

00:23:33:13 I was looking at item 22, buried 2 inch diameter conduit. Seeing a little bit of a misbalance there on Skyline and Mesa compared to everyone else. 

00:23:45:02 So again, so what we're looking for then, so at $8, I mean, 2 inch diameter conduit, I mean, that's-- that we said there was 1,000 feet. We had good numbers on it. That's just what they thought it was going to be. So then you would look elsewhere. So you're saying maybe that's low. So did they did they counterbalance somewhere else to make money up front? 

00:24:10:19 And again, we didn't see anything. And one of the takeaways here is just because the engineer's estimate and all the other numbers, the unit prices, are different and widely variable doesn't make it unbalanced. Doesn't make it penny bidding. So hold on a second, David. I think I had-- Willow were you? 

00:24:29:18 No, I rethought that. 

00:24:32:19 Would it be the control valve vault 13. 

00:24:35:18 Number 13, OK. So we said 30. I mean, they're all about the same, really, and expensive. That was a particularly expensive-- and one way to-- one sort of the-- one value of the engineers estimate is, again, we aren't penny bidding or unbalanced bidding when we put together an engineer's estimate. So if they're in the ballpark, pretty close, if they're similarly seeing an expense, then you're OK with that. 

00:25:04:13 And so number 13, we're at $30k, lowest-- the low bidder was $40, $40, you work away across. Particularly, what you're looking for-- I mean, just looking at Skyline, because they were the low bid. So I wouldn't-- Jade had one. 

00:25:21:26 Oh, I was going to say Stanley contracting seems a little odd with all of their numbers. They're all weird dollar values. And then you get to a very large amount, I guess not large. 

00:25:34:09 Yeah, you know, I honestly never quite-- I mean, although actually, if you look at the prices, I mean this is one, there's not a huge $872 to $928, not a huge range. I mean, they-- however, I've never been on the side of putting together an actual bid for construction. I've never worked for a general contractor. I mean, yeah, their numbers were-- I think one thing you would do is you would look at your two low bidders, thank goodness we didn't end up with low bid, change order low bid king there. 

00:26:15:28 We were very grateful for that. Because this is the contractor that would go out and buy equipment when he found a project that he liked, and then sell it when the project was over, and was a true pleasure to work with. But you just get these-- you get-- oddities doesn't mean unbalanced or penny bidding. You're always going to see variable. And different contractors are going to employ different approaches. 

00:26:41:19 And there might be-- it might be subtle, but it's been really hard to prove. It would have to be-- and when we look at a case study, it would have to be really egregious. So one more. 

00:26:52:25 I was just looking at line 15, because Mesa's, their estimate was $15,000, and then the next highest was $3. 

00:27:00:29 Yeah, that was abandonment. I mean, that does seem like a lot of money. And so-- but you think about, OK, I'm trying to remember where the phasing-- that was later stages of the project. So one of the things we're looking for in unbalanced is have they-- are they trying to get a lot of money upfront at the initial part of the project, and then balanced their bid to be competitive by lowering a price elsewhere. 

00:27:35:01 So really, they're getting a whole bunch of money out of item 4, whatever it might be. But really, that's going to cover the cost of item 12, which comes later in the project, different strategies. So I will tell you, I've never seen it. I've never had it happen. I've never seen an unbalanced bid. I've never seen penny bidding. 

00:27:55:25 I don't know that it's really actively pursued by contractors. But if we go back to the slides, so it's just good, because it's-- one of the things is looking at-- you're looking at these, you've got a couple of bid tabs now. You've got that one and the one I gave the other day on the pumping station. 

00:28:15:10 You can see the difference in prices. It's just-- they just are. And you don't know what's going on behind the scenes. And as long as-- one of the things you do look for in pricing, particularly for the low bid, is it-- are there particular items that are significantly lower than everybody else and the engineer's estimate? 

00:28:39:03 Again, because we're really trying to think through. We're not trying to get creative in the engineer's estimate to get some frontend money, or move things around, or whatever, look at the quantities. We're trying to literally predict what we think it's going to cost. And so if you get some that are significantly different, that could be a red flag. 

00:29:03:26 But the red flag in that case is maybe they don't understand what the work entails. That would be a red flag, if they've really underbid something that you have illustrated and described in the drawings, and in the specs, it is really significant, whether it's equipment, I mean, whatever it might be, that would be a red flag. 

00:29:29:22 So if we kind of go back to the slides here. So this is one, this was several-- obviously, several years ago. OK, this was in Boston. We had two bidders, $18 million, $20 million. The low bid had this literal penny bidding, a penny per square foot for temporary sheeting. And again, just as we've got in our instructions, the bidders commission could reject an unbalanced. 

00:30:02:25 Boston went ahead, low bid, I mean, $2 million, that's not chump change. I mean, but then the Department of Labor and Industry for the state said the bid had to be rejected. Said it was-- they did find, though, so what was interesting is they said it's unbalanced. It's not unbalanced-- because there was no-- when you see a penny, you've got to go look for a counterbalance, because it's somewhere there. 

00:30:33:15 And there was no counterbalance. But the Department of Labor and industry had a policy, that word's key, had a policy against penny bidding. And said that's not realistic. I mean, you can't accept that, because that's not really what it's going to cost. Boston said whatever the words they chose and moved forward, right? Went to a judge, judge determined that the low bid was not unbalanced or front loaded. 

00:31:08:01 Judge concluded that Boston could not move forward with that. Again, we're talking $2 million, that's not a small amount of money. So it goes to a higher court, because that's what happens. And the higher court is ruling over policy. DLI can establish policy. They can't make rules. 

00:31:27:17 So a policy and a rule are different. Legislature in this case hadn't given them that authority. All the rules come from the legislature. And then the departments enforce the codified rules. And ultimately, the judge said, look, that's fine, that's your policy. But it's a policy. It's not a rule. 

00:31:51:13 Another appeal, claimed the judge had the power to prohibit penny bidding. And ultimately in the end, low bidder was just more creative, saw an opportunity. Wasn't unbalanced, wasn't frontend loaded. So those would be grounds for rejection. She had a more creative contractor, didn't think that temporary sheeting was going to be all that important somehow. 

00:32:17:14 Maybe they'd factored it in elsewhere, but it wasn't obvious. And DLI and second low bidder had confused equal footing, which is a principle in contract law, with creative bidding to all bidders. So everybody had an equal footing. One got more creative. Why are you penalizing the more creative bidder? 

00:32:37:29 And you could almost apply this case study here to the Lewiston situation. You had three contractors that were more creative, and it would have saved the city money. Unfortunately, the city hadn't properly-- or the engineer hadn't properly developed the bid form. So yeah, I've never seen it. Looking for-- I don't know how many of these are actually out there. But it's good to know. 

00:33:06:09 I mean, it's one thing. I mean, we do scrutinize prices. We look for things that could be a red flag, that we think could cause problems should the contract be executed. And I think one more important one is-- I mean, the pricing is important. So it can be-- I wouldn't call it necessarily penny bidding, but low bidding, or underappreciating the required work, not that they have an unbalanced or loaded elsewhere, they just didn't actually understand the work. Those are the things that we look for. OK, Logan. 

00:33:41:12 Talk about frontend loading. Is there anything in the sense of backend loading? 

00:33:46:25 No, I mean, I've not heard of it. A frontend would make-- I think, the things where contractors are-- they're looking to exploit, perhaps, is underestimate of quantities, which is why we have to get our-- if we're going to do unit price, we need to measure super careful and very clearly articulate how measurement and payment. And then get it right in the bid form. 

00:34:10:05 Because if we don't, a contractor might see that, they might jack up that price, but it's a low. It's a low quantity. But then when the real quantities come into play, boom, lots of money. But backend, I don't think there's really any-- I don't know why a contractor would necessarily do that. They're looking more for exploiting weaknesses in the bid form, and maybe getting some money upfront so they're less up bank during the project. Other questions? Yeah, Alex. 

00:34:43:14 You may have hit on this. But how-- how are items-- in a lump sum contract, how is a contractor paid off of that? They're not going to get it all at once, I know they're going to slowly stagger it. But the unit price is pretty obvious. However much you put in, you keep quantities, you pay them that amount. But lump sum, I'm a little unsure how they get paid. 

00:35:05:29 How does a contractor get paid under a lump sum contract? Excellent question. It's percent completion. It's a judgment call. But we'll see that. So we'll get to that later as well. But yeah, no, good point. I mean, how do you pay? You pay on percent complete. But the-- it's going to be subjective. It's going to be subjective. And the engineer has some weight in that. 

00:35:32:17 Yeah, I mean, I much prefer unit price contracts for so many reasons. I mean, it's just-- one is it gives you the flexibility to add work if the bids come in low, which nowadays, they really aren't. But there was a time when they did. And you could add work to a project. And then it's so much easier to pay. And you're not getting into disputes. 

00:35:59:11 OK, let's see where we are here. So different type of contract, so we've talked about lump sum. We've talked about unit price. This would be a cost plus contract. This is where you're paying for the actual work. You're paying-- you're actually paying for the labor, the materials, and supplies, plus a percentage or fee, that fee basis, that's the overhead and profit. 

00:36:23:25 So you're actually pulling them apart. We're going to just pay you for the work. And then we're going to have an agreed upon percentage or fixed fee that's your overhead and profit, rarely used, more emergency project. Where you just say we need to get a contractor on board. We had-- during some flooding in '96, there were-- the city of Amity in Oregon, the whole Yamhill water-- or Yamhill River went over its banks. 

00:36:52:24 And ended up totally silting in their raw water intake for their water plant. And so we had to bring a contractor on board in that situation. And we didn't have time to bid it. We were able to just hire a contractor, under a structure like this. We'll just pay you for the work you do. And then we have an agreed upon percentage or fixed fee. So those types of-- that's where you would use it. 

00:37:17:00 You could use it in difficult or unique projects. We had-- that was actually the same contractor. It was the city of Hood River. It was actually design build project. So the engineer was partnered with the contractor. But sort of similar, we'll talk about that more later, and then I have a guest lecturer at some point later in the semester who's very knowledgeable in these alternate delivery methods, including design build. 

00:37:46:19 But this was a situation where we had springs that the city had developed back in the early 1900s. We knew next to nothing about what we were going to see. And so as the-- we were doing preliminary design with the contractor for the city. And then there came a point where then the owner was able to work with the contractor to negotiate a sort of a similar cost plus type contract, given all the uncertainties. 

00:38:18:29 Because we didn't know what to expect. And thankfully, one of our myriad designs is what we used. Not the one we put on the drawings, but one of the ones we'd actually talked through, which was useful. It was actually a lot of fun. If you can get into a design build project, that can be with a contractor, it can be a lot of fun. 

00:38:39:04 OK, so here's-- I'm going to just talk about-- and we'll go look at some general conditions here to walk through this. So again, an engineer friend of mine had a project up in northern Idaho. They were-- it was a sewer project. It was a cured in place. So this was a special-- very specialized work. It was a contractor that would-- it was-- I don't know if it was Insituform or Gelco. I don't know who or what the contractor was, but-- who it was. 

00:39:08:28 But they're pulling a liner. They're repairing a pipe in place, cured in place. So they're not excavating or removing the pipe. They're not-- they're not pipe bursting, pulling a new one in. They're just pulling a liner in. And then they actually can-- it's a particular chemical structure. And then they heat it up. And it creates a whole new pipe, if you will. 

00:39:33:19 So they went to the contractor and said, hey, we'd really like to add some work to the project. But the project didn't have line item unit price on some excavation. So they had the contractor, again, was not an excavation project, had to go find a sub. And then they submitted a price. So they found a subcontractor who can do the excavation. They went, submitted the price to the owner. 

00:40:01:25 And then the general contractor passed along the cost, so it was a cost plus type contract, but a 20% markup for overhead, admin, profit, et cetera. So the question is, do you just-- do accept it? Do you pay them the 20%? So this is where we need to go-- I'll let you get to general condition. 

00:40:24:19 Actually, we'll go to Article 12 in the General Conditions, which is change of contract price or change in contract times. I'll let you find that. So this is page EJCDC C-700 page 46, at the bottom. So here's one. I mean, this is a real situation. And this engineer had to navigate it. 

00:40:56:20 So if we look at 12.01.C, 12.01.C.1.a, that's where you start. So here's a change of contract price. That's what this was. You were asking for a change order. The contract may be only changed by a change order. So that's a formal document. It's in capital letters purposely. It's defined in Article 1 of the General Conditions. 

00:41:23:14 Contractor's fee, that's what we're speaking of here, the contractor's fee for overhead and profit shall be determined as follows. So the general conditions in the contract that have been signed for allows-- I mean, it gives you clarity on how to deal with that. You can be a mutually accepted fixed fee, which wasn't the case. 

00:41:48:12 If a fixed fee is not agreed upon, then a fee based on the following percentages of the various portions of the cost of the work for costs incurred under 11.01.A, et cetera, shall be 15%. Well that's not 20%. Contractor just did 20%, didn't adhere to the contract at all, 15%. 

00:42:14:09 But then if-- so if we look at-- so C-- that's C.2.a, right? But then, it gets a little more gray. For costs incurred under 11.01.A 1 and 2, it's 15%. But if it's A.3, it's 5%. So it's either 15 or 5. But now, we have to go look at 11.01.A.1, A.2, and A.3. 

00:42:54:20 So A.1 gets into some great-- spells it out in great detail, the cost of the work means the sum of all costs. You get all these-- one of the things you're seeing is it doesn't repeat. It says except as those excluded in 11.01.B. So you have to look at 11.01.B. So you're doing a little bouncing around, which can be frustrating. 

00:43:20:11 When the value of any work, covered by a change order or a claim, which is different than a change order, is based on the cost of the work, the cost to be reimbursed will only be those-- will be only those additional or incremental costs required because of the change because of the event giving rise to the claim, except as otherwise may be agreed in writing, such costs should be the amounts no higher than those prevailing in the local. 

00:43:49:18 That's subjective, right? Shouldn't be those that it would not exceed prevailing. So we've got payroll costs. I'm going to scroll through this really quickly. A whole bunch of them, costs excluded. But then C, so here's-- when all the work performed is performed on the basis of cost plus, so we have two different ones-- so it was 15% if it was A.1, which is payroll costs, what was it? A.1 and 2, I forget? Or yeah, A.1 and 2, right? 

00:44:32:28 So it's 15% on top of what the contractor is proposing if what the contractor is proposing is payroll costs and materials and equipment, furnished and incorporated into the work. So that's 15%. But then, if we look at C, as set forth in the agreement. So all sorts of-- I mean, it's clear but not clear. 

00:45:06:28 So again, the first part deals with payroll costs, materials. So the challenge that he faced was what do you do, right? Clearly, it's not 20%. Right from the get go, it's not 20%. But is it 15%? Or is it 5%? 

00:45:26:12 His conclusion, based on the contract that was in play, is that the general conditions indicated 5% was the right number. Now, the contractor is not required to execute the change order. You invite-- you ask for a change order. You ask them to perform extra work. But if the price isn't agreed upon, it doesn't happen. So nothing is-- nothing is requiring the contractor to accept the terms. 

00:45:54:05 Of course, the contractor in this case wasn't super thrilled about 5% for whatever reason, especially, it wasn't really their expertise. They had to really-- they had to bring on a sub. They'd have liked to get a little bit more out of the deal. And they countered with 12%. And they ultimately did settle on 12%. 

00:46:16:16 So a little bit of negotiation, owners involved on that. They decided there would be some direct coordination work in the field and the office. So direct payroll costs, so there would have been-- it would have required payroll costs of the general contractor to, actively in the field, oversee the work performed by the subcontractor that was being brought into play. And so that would lean you toward the 15%. 

00:46:47:06 But 15% didn't feel quite right. And they settled on 12%. So you could see where, depending on what you're asking, it could go either way. But the nice thing is, at least the general conditions cover us. And they allow for a discussion and a negotiation. We're not held to whatever dollar amount the general contractor wants to do. 

00:47:18:11 And so I'll just-- I'm going to read this verbatim. So this is what he said. The rest of the story here that you won't find guidance on in EJCDC, and that I'm sure you're familiar with, from my experiences, is the give and take of construction and the need to pick battles. He goes, I didn't want to be too much of a stickler, because-- I'm going to make sure he's not using inappropriate words, anything I don't want to put on tape, didn't want to be too much of a stickler, because they were are really helping the owner by taking on the subcontractor and agreed to do so even though they typically don't. 

00:47:54:07 Again, this was a very specialty contractor that they don't do excavation. So this was very much a favor to the owner. They'd also done some additional TV work on the lines that we thought we might add to the project that we ultimately didn't based on-- so there were some-- apparently some work that they didn't get paid for that was originally part of the bid. They weren't planning on claiming that additional work anywhere else. So even though they had some time. 

00:48:25:23 So it was ultimately in the best interest to not make this a big deal. The city didn't pay a premium for the markup. But our generals still came out with something. So ultimately, it was a win-win. So that's the general conditions give us a lot of direction. But there is still that negotiating with contractors. A win-win, you want it to be a win for the owner. 

00:48:48:04 And the other way to think about it, because again, the contractor could have said, no, it's not it's not our bailiwick. It's not in our world. We don't want to do this. So then you would have to prepare bid documents, go out to bid, bid the work, and then go through that whole process. 

00:49:05:11 So all the engineering costs, so there's a lot of extra costs that would have been incurred if the general contractor said no, we don't like your terms. We're not going to do this. So you have to factor that in, as well. What's the alternate approach? And is that more expensive? Or is this a very expedient way, and so you compromise. 

00:49:29:26 So I think those kind of stories are interesting to share. Let's see where are we at on slides? I got beeped. I'll let you look at-- I would encourage you, because I fully expect to have at least a question on the exam that speaks to 11.01, our general condition 11 and general condition 12. So you should be looking at those as you think about the exam. And we will pick up right there on Monday. Have a good weekend. 

Duration:"01:16:32.4080000"

00:00:21.550 -- OK, a couple things as we get started. The first one is we

00:00:26.828 -- have the last lab assignment.

00:00:31.670 -- And so this one is going to be a bus differential protection lab

00:00:35.349 -- and so the on campus students is pretty much going to be a

00:00:39.028 -- similar setup to what you did before. You just need to read

00:00:42.424 -- through this and then work with the TA. As far as if you're

00:00:46.103 -- going to, I think you all of you have groups that you've been

00:00:49.782 -- doing the labs with the TA. If you want to stick with those

00:00:53.461 -- groups in those times. If you wanted to negotiate a different

00:00:56.574 -- time, then you just need to communicate with him about that.

00:01:02.040 -- Until you have a system and you're going to look at fault

00:01:05.628 -- at a couple of different places, this is actually

00:01:08.319 -- should be a little bit shorter than the last, quite a bit

00:01:11.907 -- shorter than the last lab.

00:01:14.920 -- And so you're really just going to look at several

00:01:17.710 -- different cases.

00:01:19.710 -- Look at the behavior with this.

00:01:22.940 -- The Engineering Outreach Lab is going to be similar.

00:01:26.730 -- So this is just the description of the entering outreach lab.

00:01:31.580 -- And so it's a little bit more complicated system, but it's

00:01:34.616 -- still the same basic idea.

00:01:36.980 -- And also you have some information about the CT

00:01:40.410 -- ratio that's was used for this.

00:01:44.480 -- And then this is using that.

00:01:47.590 -- Relay model that the differential relay model we

00:01:50.462 -- talked about. So again this is a low impedance restrained

00:01:54.052 -- differential element, so it's not. It's not a high

00:01:57.283 -- impedance differential element.

00:02:01.530 -- If anyone has fair time and wants to create their own

00:02:05.369 -- creative all the create this, it wouldn't be that hard to

00:02:09.208 -- create a lab for the restraint for the high impedance

00:02:12.698 -- differential elements. We just haven't had a chance to put

00:02:16.188 -- together the simulation files.

00:02:18.800 -- So anyway, it's the same idea you read in the data files.

00:02:24.470 -- Very similar to the handout that we talked about with the lecture

00:02:28.310 -- last week. All of this stuff we're reading the comtrade file,

00:02:31.830 -- and so where this really starts to differ a little bit is

00:02:35.670 -- towards the end of it. Once we've got the phasers, so we've

00:02:39.510 -- got the things where we're looking at the voltages in the

00:02:43.030 -- currents, and then we have the operating restraint current, and

00:02:46.230 -- so one thing that's different from the handout before is now

00:02:50.070 -- the. In this case, there's no.

00:02:53.660 -- Nothing where you put in a multiplier to imitate

00:02:56.297 -- saturation. The simulation data that you're using for this now

00:02:59.227 -- actually has saturation in it.

00:03:01.850 -- And the case is that you'll be doing for the on campus

00:03:05.450 -- students in the lab. You're actually going to be doing

00:03:08.450 -- these with an RTS simulation instead of using the model

00:03:11.450 -- power system, and so that the RTS will have setae. Models

00:03:14.750 -- that include saturation, but you're still going to be

00:03:17.450 -- setting the actual physical relay.

00:03:21.310 -- And then one of the things that this is going to show is the

00:03:25.664 -- basically the how they operate. Quantity changes. So basically

00:03:28.463 -- as it reads through samples, this thing is moving and then it

00:03:32.195 -- works its way up and then it has some final value it goes to and

00:03:36.860 -- so you can as you look at these different cases once you enter

00:03:40.903 -- the slope setting you can actually look at a little bit

00:03:44.324 -- how the how the value evolves and when you look at the case

00:03:48.367 -- with the saturation you can actually see how it.

00:03:51.300 -- Now the saturation changes what it's what the relay

00:03:54.171 -- element is seeing too, and so this was a case for an

00:03:57.999 -- internal fault, so it grows quickly.

00:04:02.860 -- So any questions about that?

00:04:09.170 -- Hey are there any questions from the last lecture?

00:04:12.680 -- Yeah, so in the last lecture when you talk about the high

00:04:16.832 -- impedance plus differential protection, you mentioned that

00:04:19.254 -- for an external fault. Once one of the see T starts to saturate

00:04:23.752 -- it will dive deeper into saturation, right? So my

00:04:26.866 -- question is how will that?

00:04:29.160 -- To how will that city begin to saturate? Like because?

00:04:33.660 -- The currents are all balanced, right? I mean based on the

00:04:37.972 -- culture of slow, so part of it's too far into this into the

00:04:43.460 -- hand out so.

00:04:47.760 -- That's the internal fault. So for the external fault part of

00:04:51.148 -- it's going to be the case that.

00:04:54.690 -- We've got this one. This is 1 heck external fault, right? So

00:04:58.338 -- this is seeing the current from all of the other feeders or

00:05:01.986 -- other lines going through it, and so depending on what the

00:05:05.330 -- burden is for this one.

00:05:07.800 -- Oh that 'cause there's going to be?

00:05:12.460 -- The relay and the and some of the winding resistance is going

00:05:16.084 -- to be dominant. Burden that affect saturation in this one in

00:05:19.406 -- a lot of ways.

00:05:21.210 -- So if this one, if there's a fault with a lot of DC offset,

00:05:25.088 -- especially then this one is going to start to saturate.

00:05:27.858 -- 'cause this is seeing the most current. I thought there is only

00:05:31.182 -- one button then that's the one at the end. Well, remember that

00:05:34.506 -- the burden and we look at ACT when we look at burden.

00:05:40.260 -- Mr Lead wire.

00:05:48.530 -- So the first thing we're going to have is the CT winding

00:05:51.338 -- resistance. And it's so. So in this case the Siti

00:05:54.649 -- winding resistance is going to be the most significant

00:05:57.088 -- one, because once we get to the terminals of the see T.

00:06:07.470 -- We're basically connecting each of the CTS.

00:06:12.750 -- In parallel on the secondary side, right and then once

00:06:16.320 -- they once we have this parallel combination, then

00:06:19.176 -- that's going. Then we have the rest of the lead wire.

00:06:25.200 -- And we have the relay out here.

00:06:30.540 -- But there's the secondary current on the secondary

00:06:33.404 -- winding, and the CT is still going to see.

00:06:37.340 -- All that current, right? The current when they sum

00:06:40.328 -- to 0 between.

00:06:44.320 -- We put in a third CT just to kind of.

00:06:49.030 -- Illustrate this a little bit more.

00:06:58.680 -- When I talk about connecting them together right, this is

00:07:01.940 -- where they sum to 0, right? So if it's if it's an

00:07:05.852 -- external fault.

00:07:12.350 -- So let's say that this is the one with.

00:07:18.870 -- The external fault, right? So that's going to have.

00:07:23.100 -- Let's say we have current going this way and this one. Each of

00:07:26.948 -- these are going to have their share feeding it right, so this

00:07:30.500 -- one is going to be the sum of this plus this and so at this

00:07:34.940 -- point here. They're going to sum

00:07:37.224 -- to 0. But this one, each one of these is going to have its own

00:07:42.072 -- fault current share the fault current, it's it's

00:07:44.628 -- carrying. It's going to go

00:07:46.048 -- through this resistance. And so basically what's going to

00:07:49.888 -- drive that start driving in this one in the saturation is

00:07:53.936 -- going to be a combination of the voltage drop across this

00:07:57.984 -- plus the ACE asymmetric current due to the DC offset.

00:08:03.130 -- Remember that as we talked about with on the BH

00:08:07.030 -- characteristic, the DC offset is shifting you in One

00:08:10.540 -- Direction and the BH characteristic.

00:08:17.450 -- And so when we look at this.

00:08:21.080 -- So under normal conditions.

00:08:23.650 -- It's going to be doing something like this, right? And

00:08:26.760 -- if we have a fault with no set without significant saturation?

00:08:31.480 -- It's going to be doing some like this, and so if we have well

00:08:36.324 -- size CTS we may only see behavior that looks like this.

00:08:40.730 -- But for a bus situation, sometimes it's hard to get

00:08:44.380 -- around that, but if we add.

00:08:47.510 -- The.

00:08:52.590 -- The DC offset.

00:08:54.830 -- I did not draw that very well, sorry. So we may start out with

00:09:00.248 -- something like this. Then the

00:09:02.183 -- next cycle. It's going to be working like this and it's going

00:09:06.480 -- to be following that DC offset, so it's going to push it into

00:09:10.302 -- saturation. Discuss. The flux loops are being pushed this way

00:09:13.242 -- by the DC offset.

00:09:15.870 -- And in some cases with a combination of the of a large

00:09:20.286 -- current and going through this resistance in a DC

00:09:23.598 -- offset, this one may start to go into saturation an.

00:09:29.390 -- Lessina cycle.

00:09:31.800 -- Possibly quite a bit less in the cycle.

00:09:35.790 -- And so that's why that's why even though you on the surface,

00:09:39.606 -- you would say that there shouldn't be much voltage across

00:09:42.786 -- this, because these current sum to zero and the voltage drop

00:09:46.284 -- across this should normally be negligible. But what's going to

00:09:49.464 -- happen is that the combination of that fault current going

00:09:52.644 -- through this winding resistance and the DC offset starts this

00:09:55.824 -- one into saturation. And then that mismatch current through.

00:10:00.130 -- That saturation goes through this, and because of that

00:10:03.577 -- compensating resistor that's going to drive this voltage up.

00:10:08.730 -- But because this is the one that's already starting to

00:10:12.290 -- saturate and has a lower impedance than it's, it's

00:10:15.494 -- going to tend to make this voltage collapse and keep

00:10:19.054 -- these from rising.

00:10:28.560 -- Like I said, it's not. That's a very good question. 'cause it's

00:10:32.280 -- there's a lot of things that are not intuitively obvious when we

00:10:36.000 -- look at the high impedance bus

00:10:37.860 -- differential. Because we're basically using something that's

00:10:42.662 -- inherently nonlinear to work.

00:10:57.580 -- Any other questions for my son?

00:11:06.790 -- OK, so then we're going to start on. Next, we're going to start

00:11:11.223 -- talking bout transformer protection and I talked to I did

00:11:14.633 -- a very quick introduction to some of the some of the issues

00:11:18.725 -- and the difference.

00:11:20.960 -- Things were gonna talk about.

00:11:21.870 -- We're going to talk about. Fall protection of the

00:11:25.190 -- transformer itself for faults inside the transformer.

00:11:29.680 -- And then we're also going to look at protecting the

00:11:32.850 -- transformer, firm external conditions, and

00:11:34.435 -- this can include faults external to the

00:11:36.654 -- transformer. Boy, the transformer is carrying

00:11:38.556 -- the fault currents that goes that go to it.

00:11:47.680 -- And then there are Transformers introduce a number of unique

00:11:51.640 -- challenges that we'll talk about as we go through this.

00:11:56.550 -- So in some ways it will start out looking at a concept similar

00:12:01.308 -- to what we looked at with the bus protection. So we're going

00:12:05.700 -- to a lot of the internal fault protection for Transformers.

00:12:09.360 -- Starts with the idea of restrained low impedance

00:12:12.288 -- differential element, so it's kind of build time. We start. I

00:12:16.314 -- started with the bus protection.

00:12:29.630 -- And so one of the things that the bear in mind as we talk

00:12:35.748 -- about transformer protection is when we talk about bus

00:12:39.681 -- protection. Fast protection has a bus fault or misoperation

00:12:43.614 -- where a bus gets tripped when it shouldn't can have very severe

00:12:48.858 -- operational. Consequences for our power system. So bus faults

00:12:52.556 -- are actually fairly rare.

00:12:54.760 -- Fat faults that cause were and the bigger concern is as

00:12:59.028 -- generally going to be external faults that caused the bus

00:13:02.908 -- protection to miss operate.

00:13:06.160 -- And so that's why the restrained differential element, the high

00:13:09.640 -- impedance differential element, have so there so much efforts

00:13:12.772 -- gone into developing and optimizing those at the relay

00:13:15.904 -- vendors is because they are very high consequences operationally

00:13:19.036 -- to the system in the short term.

00:13:24.130 -- Transformer failures, on the other hand.

00:13:44.050 -- Can have longer time consequences.

00:13:54.730 -- And that's because there are longer replacement times.

00:13:59.700 -- And in most cases, if an internal fault happens in a

00:14:04.560 -- transformer.

00:14:06.370 -- There is a good chance that it's going to evolve to the point

00:14:10.348 -- where it's not something that's very simply repaired. In some

00:14:13.408 -- cases there are still a number of cases where they're caught

00:14:16.774 -- fast enough, or it could be repaired simply, but if it gets

00:14:20.446 -- to severe faults and you'll have a fire in the transformer, then

00:14:24.118 -- it can be very severe.

00:14:27.950 -- And so there are a number of things. The number of strategies

00:14:32.750 -- that try to minimize the impact of transformer faults.

00:14:49.760 -- So one of the big ones is finding ways to reduce the

00:14:53.252 -- likelihood of them happening.

00:15:05.320 -- And so part of what a lot of this comes down to is.

00:15:10.900 -- Track external events.

00:15:31.620 -- And it's really the life of the installation. That's a

00:15:34.010 -- big issue.

00:15:35.620 -- So one of the things that I mentioned is that we have two

00:15:39.741 -- directions. We're gonna go to, and they actually are related to

00:15:43.228 -- each other. So one of the big things that is a has a

00:15:47.349 -- consequence for Transformers is.

00:16:06.620 -- Meeting of the installation will have a big impact on

00:16:09.700 -- how the life or how long that installation is going

00:16:12.780 -- to be good.

00:16:23.030 -- Transient overvoltages is another another issue.

00:16:44.770 -- So what are some of the things that are going to

00:16:47.168 -- cause a transformer? Cause heating in a transformer?

00:16:51.350 -- So let's think about a transformer for a second

00:16:53.690 -- we have.

00:16:56.970 -- So I'm just going to draw a single phase core.

00:17:01.570 -- So as we've talked about where we have a single phase core

00:17:05.410 -- and have the low voltage winding on the inside, an will

00:17:08.930 -- have a higher voltage winding wrapped around the outside of

00:17:12.130 -- it, right? And then we'll take those out to the bushings.

00:17:16.690 -- And as I mentioned earlier, we don't. You don't see a

00:17:20.397 -- transformer core just sitting out open in the air, right?

00:17:24.360 -- And so usually this is going to be.

00:17:32.140 -- In a tank.

00:17:37.710 -- Anna's tank is going to be.

00:17:45.730 -- Filled with oil, right? So usually it's going to be some

00:17:48.271 -- sort of a dielectric oil.

00:18:02.600 -- Is also used as a coolant.

00:18:08.650 -- And so you may look at a name plate for a transformer, an it

00:18:13.914 -- may say that you have a transformer that's rated at 15

00:18:18.050 -- MVA, 20 MVA.

00:18:20.300 -- 25 NBA

00:18:23.920 -- so why would why would there be 3 MVA ratings for the

00:18:27.712 -- same transformer?

00:18:34.120 -- Different cooling stages. It's different cooling stages, so

00:18:37.424 -- this is going to be.

00:18:40.720 -- Basically, entirely passive cooling.

00:18:45.100 -- So there is going to be there will be radiator fins or on the

00:18:49.510 -- side of this case on the side of

00:18:52.030 -- that tank. This is going to be.

00:19:03.070 -- Going to be pumps used to circulate oil to cool the

00:19:06.029 -- transformer or cool the oil so it's going to circulate because

00:19:08.988 -- there are going to be.

00:19:11.010 -- Different spots in the winding that are hot spots said certain

00:19:14.156 -- certain points are going to be

00:19:15.872 -- hotter than others. And so if you don't circulate the coolant,

00:19:19.424 -- there will be a little bit of natural convection, but you're

00:19:22.262 -- going to. Those hot spots are not going to be cooled as well.

00:19:26.580 -- And then this is going to be pumps.

00:19:31.090 -- Plus

00:19:32.920 -- running cooling fans that are blowing error basically across

00:19:36.997 -- the radiator so that the radiator works more efficiently.

00:19:45.810 -- So depending in some cases people will just run these

00:19:49.220 -- all the time. In some cases they'll based on the load

00:19:52.971 -- conditions, they'll start and stop this equipment.

00:19:56.890 -- And if you have a transformer that's always lightly loaded,

00:19:59.290 -- they may not. Run it as. Run to run them very much at all.

00:20:15.100 -- So other things that could cause heating.

00:20:23.270 -- So I want to be carrying harmonic currents.

00:20:38.460 -- Do you know external loads?

00:20:48.380 -- So for example, if we have a transformer that one of

00:20:52.153 -- the loads.

00:20:54.110 -- Is.

00:20:59.190 -- A dialed dialed rectifier.

00:21:04.740 -- And then we have a voltage source converter.

00:21:09.580 -- Anyway, have an induction motor.

00:21:17.060 -- If.

00:21:19.260 -- This doesn't have any compensation.

00:21:28.570 -- The current strong by this drive are going to look

00:21:30.800 -- something like this.

00:21:34.920 -- And so this is going to have 5, seven, 1113 and

00:21:39.463 -- basically multiples of 6 plus or minus one.

00:21:47.670 -- Is there going to have other loads here? But this

00:21:50.100 -- transformer is going to be carrying this current plus

00:21:52.287 -- whatever loads are here.

00:21:57.000 -- And carrying those harmonic currents increases Eddy current

00:22:00.808 -- losses in the transformer core.

00:22:04.480 -- And so that the transformer is going to run hotter.

00:22:23.280 -- And so they actually you can actually get.

00:22:27.470 -- K factor rated.

00:22:35.920 -- So basically these K factors are more of a derating factor.

00:22:41.170 -- And so if you have, if you know you're going to be supplying

00:22:45.642 -- harmonic loads, you can buy a transformer that has basically

00:22:49.082 -- an extra factor in its MVA rating to be able to deal with

00:22:53.554 -- harmonics. If you're not, if you don't have a transformer

00:22:58.028 -- that has any K rating an you start supplying harmonics,

00:23:01.848 -- then usually you can. There's there are formulas from the

00:23:05.668 -- IEEE standards that talked about how you derate the

00:23:09.106 -- transformer, so instead of being a 15 MVA transformer, it

00:23:12.926 -- may actually be a 12 MVA transformer due to the extra

00:23:17.128 -- heating from the harmonics.

00:23:19.820 -- And so when someone buys a transformer, usually you're.

00:23:24.470 -- Part of the data for when you sign the contract with the

00:23:28.019 -- supplier and stuff like that is saying well, this is. This has a

00:23:31.568 -- 30 year design life for this as a 25 year design life.

00:23:35.850 -- If you routinely overheat the transformer, you may take years

00:23:39.750 -- off of that life.

00:23:41.870 -- So we had an outreach student awhile back that worked at an

00:23:46.334 -- industrial facility that was basically with zinc smelter.

00:23:50.010 -- And so they had a lot of very large rectifier loads and so

00:23:54.924 -- they had Trent. They bought Transformers that had.

00:23:59.340 -- 30 year old designlife

00:24:01.880 -- Then they push them kind of right. It may be a slightly

00:24:07.520 -- beyond their NBA ratings.

00:24:10.240 -- And then they gave this heavy harmonic loading. So they

00:24:13.060 -- lasted about 10 years.

00:24:18.790 -- An that fit and when I say lasted about 10 years, they had

00:24:24.237 -- a fault, and so if I did so by heating the insulation, you end

00:24:30.103 -- up causing the you decrease the lifespan of the installation and

00:24:34.712 -- your moral an it's more likely to fail by having our fault. And

00:24:40.159 -- so that's why this external event, external condition stuff

00:24:44.349 -- matters from the from the transformer Protection POV.

00:24:52.670 -- So transformer protection will usually track the loading on a

00:24:56.900 -- transformer an if the transformer is overloaded, and

00:25:00.284 -- then there are formulas you can use to figure out how much

00:25:05.360 -- that's affected the life.

00:25:10.980 -- And so some other things that will go into this are going

00:25:13.776 -- to be over excitation.

00:25:23.720 -- So on a transformer over excitation basically means

00:25:26.400 -- a steady state.

00:25:34.200 -- However, voltage that means you're partially saturating.

00:25:57.850 -- Angene why the transformer is going to produce more

00:26:01.478 -- harmonics because of this? Because this is a steady state

00:26:05.438 -- sinusoidal condition, these will be only odd harmonics.

00:26:11.110 -- And often the 5th harmonic is usually going to be the one

00:26:14.674 -- that's used as sort of the main detection detector for that.

00:26:21.140 -- But again, because you're saturating the core.

00:26:26.070 -- What does that? What does it mean when you saturate

00:26:28.960 -- the core more deeply?

00:26:35.050 -- More excited, you have more expectations, well over

00:26:37.938 -- expectations. We have more expectation right? But what

00:26:40.826 -- losses go up?

00:26:44.480 -- The winding losses go up, or so we're going to increase

00:26:50.200 -- hysteresis losses.

00:26:54.180 -- Remember, hysteresis losses are basically proportional to

00:26:56.672 -- the area of the hysteresis loop it follows, so if you're

00:27:00.588 -- over exciting the transformer, your loop has a bigger bigger

00:27:04.148 -- area, so the losses are going to be higher.

00:27:24.780 -- Another one that's a big factor are through faults, which means

00:27:29.345 -- that the transformer.

00:27:52.160 -- So basically, one of the things that also gets tracked is how

00:27:56.120 -- many, how many faults is this transformer supplied? What is

00:27:59.420 -- the magnitude of the fault

00:28:01.070 -- current bin? Because. Oh through fault can cause very substantial

00:28:05.092 -- heating. It may not. It's not going to last very long, but

00:28:08.764 -- it's going to take a long time. It's going to take awhile quite

00:28:12.742 -- awhile for the transformer to cool down from that.

00:28:37.870 -- So even frequent large motor starting or if the transformer

00:28:42.020 -- is supplying current to energize other Transformers.

00:28:48.500 -- So for example when.

00:28:53.020 -- I think their procedures have changed a little bit, but at

00:28:56.771 -- Grand Coulee there's a pumped hydro storage facility that

00:28:59.840 -- has very large synchronous Motors. They generally only

00:29:02.568 -- start those Motors once a day because the thermal shock on

00:29:06.319 -- the Motors every time they start them is so much that

00:29:10.070 -- they can't start them more often.

00:29:14.040 -- They redid that facility.

00:29:17.610 -- And within the last.

00:29:20.130 -- Eight years, so I think they've redone it, so

00:29:23.019 -- it's not quite as harsh.

00:29:26.260 -- But so basically all of these things get tracked.

00:29:45.910 -- They predict lifespan loss and we're going to. We're going to

00:29:48.814 -- come back and talk about the over some of these issues and

00:29:51.982 -- how and how this factors into the transformer protection later

00:29:54.622 -- in the course. I want to talk about internal faults. First,

00:29:57.526 -- we're going to come back to

00:29:59.110 -- this. That a good resource for this. Our textbook does a pretty

00:30:03.945 -- good job with this, but also the IEEE 30 C 3791.

00:30:08.770 -- Also another good one for this and or there's some

00:30:11.730 -- other references. We'll talk about a little bit later.

00:30:21.310 -- And So what I want to start talking about is now protection.

00:30:27.370 -- For internal faults.

00:30:33.170 -- And will be going through this over the next couple

00:30:35.320 -- of lectures.

00:30:45.020 -- And so I guess that's one other sort of structural

00:30:47.990 -- thing. When we look at.

00:30:51.370 -- Large Transformers again.

00:31:13.460 -- I felt it evolved to the point where there's

00:31:15.485 -- a fire can cause long.

00:31:19.490 -- As I said, long repair times.

00:31:23.550 -- And so some of the things that you'll see in a substation, for

00:31:29.205 -- example for large transfer transmission substations

00:31:31.815 -- especially often you'll see single phase Transformers used,

00:31:35.295 -- and so you'll see.

00:31:40.310 -- Three single phase units, and actually they are often going

00:31:43.810 -- to be 3 winding Transformers as we talked about earlier in

00:31:47.660 -- the semester.

00:31:49.800 -- And so they're going to have their own individual tanks.

00:32:00.700 -- And when you look at the substation.

00:32:04.490 -- You'll see a wall that's been placed.

00:32:09.980 -- Between the Transformers.

00:32:13.120 -- So what's the purpose of that wall?

00:32:16.130 -- Prevent fire from cleaning, so these are.

00:32:20.530 -- Firewalls raise more of the archaic usage of the term

00:32:24.070 -- instead of the one that's now everyone uses when they talk

00:32:27.964 -- about software.

00:32:29.990 -- And so this is basically if this one has a fault, and as

00:32:34.072 -- a fire, the idea is that this is that this is going

00:32:37.840 -- to basically make it less likely for any for the heat

00:32:41.294 -- in the flames to get to this transformer, so it fails to.

00:32:49.730 -- And a lot of utilities will

00:32:52.352 -- have. A limited number of spare Transformers that they

00:32:56.392 -- can put in to replace a failed transformer.

00:33:00.350 -- So.

00:33:03.620 -- This was probably almost 15 years ago. Now there was a

00:33:08.328 -- transformer fault at a 500 kva. Think it's a 500KV substation in

00:33:13.464 -- the Southwest. An they did not have.

00:33:18.530 -- Firewalls between the single phase transformer, so they lost

00:33:22.364 -- all three phases. They had their spares close enough that it

00:33:27.050 -- actually scorched the paint off of the tanks, but they actually

00:33:31.736 -- didn't lose the spares.

00:33:36.820 -- But because they lost all three and they only had three spares,

00:33:41.392 -- then they had to scramble to try to get spares from other people.

00:33:46.345 -- And I know that one of the utilities in the northwest

00:33:50.536 -- sentence pairs and they had all sorts of issues because these

00:33:54.727 -- were 500 kva Transformers, Oran, high MVA ratings. Just

00:33:58.156 -- transporting them was difficult.

00:34:04.410 -- And I think even transporting the spares

00:34:06.867 -- took like several months.

00:34:17.190 -- So then actually one of the things that the.

00:34:21.070 -- US Department of Energy in the Department of Homeland

00:34:24.814 -- Security been working on in the last several years, is

00:34:28.974 -- basically trying to form a kind of a national database

00:34:33.134 -- of transformer spares and also trying to increase the

00:34:36.878 -- inventory of spares so that if there is something like.

00:34:42.950 -- High energy electromagnetic pulse from a nuclear weapon or a

00:34:47.550 -- major Geo Geo magnetic.

00:34:50.070 -- A disturbance for the gym geomagnetically induced currents

00:34:53.454 -- caused transformer failures that they've got something that they

00:34:57.261 -- can go to restore power in some

00:35:00.222 -- areas quickly. Relatively quickly.

00:35:05.270 -- OK, so let's now start talking a little bit more about the

00:35:08.798 -- Internal fault protection.

00:35:16.340 -- Really, the first line for this is going to be

00:35:19.390 -- differential protection.

00:35:27.600 -- So as I said, much like what we were just talking

00:35:31.285 -- about with the.

00:35:33.640 -- Boss protection for the restrained low impedance

00:35:38.078 -- differential protection.

00:35:42.100 -- So let's start out looking at a transformer that.

00:35:47.170 -- We have a YY connection.

00:35:51.350 -- And so, let's say it's.

00:35:55.330 -- 3:45 KV. 2.

00:36:00.110 -- 138 KV.

00:36:07.700 -- And so for the moment, let's just say it's a.

00:36:11.870 -- 2 winding Transformers. So we're going to have

00:36:14.102 -- three leads coming out.

00:36:34.210 -- Now I have see T is on each phase and will just look at one

00:36:38.110 -- phase for the moment.

00:36:47.100 -- And so we start out saying, OK, well, this looks a lot

00:36:50.436 -- like what we talked about when we anytime we talked

00:36:53.216 -- about differential protection. So we're going to

00:36:55.162 -- have current if we have current going this way.

00:37:02.630 -- Then we're going to have.

00:37:06.340 -- Secondary current. That's going to circulate like this, and.

00:37:12.870 -- I op should be about 0, right? That would be. That's

00:37:17.666 -- what we would expect.

00:37:23.200 -- Now, unlike the virus protection, we've got a number

00:37:27.574 -- of factors that complicate this.

00:37:44.360 -- So what do you think? Some of the complicating factors

00:37:46.660 -- might be?

00:37:49.540 -- Configuration. Well, let's say they will stick with the

00:37:53.020 -- YY for the moment.

00:37:56.010 -- If it's why Delta that, that will add, that will be the next

00:37:59.195 -- challenge, will talk about after we finish this one.

00:38:03.450 -- CD accuracy. Find CD accuracy.

00:38:07.640 -- So ciety accuracy, but there's actually something

00:38:09.831 -- before that. One is going to be the CT ratios.

00:38:37.200 -- So we may not get apart. We may not get a perfect

00:38:40.284 -- cancellation of.

00:38:42.410 -- So let's say that just for making this easier, let's say

00:38:46.172 -- that this was a 2 to one ratio.

00:38:54.770 -- So let's say that this was 500KV and this was 250KV just

00:38:58.598 -- for nice numbers. Even though the 2:50 is not something

00:39:01.788 -- you'd run across much.

00:39:04.660 -- Then we would say OK. Well then this. Let's say that

00:39:07.608 -- this is 1000 to one CT and this is going to be what?

00:39:15.290 -- Or 1000 to 5C T, and that's what would this

00:39:17.760 -- would need to be then.

00:39:24.010 -- Remember, this is.

00:39:26.200 -- Two to one is the effective voltage transformation

00:39:28.680 -- ratio, so the current goes the opposite, right?

00:39:32.170 -- So this one would need to have 500 to 5 setes.

00:39:37.110 -- So that would be one that would be an example of a

00:39:39.894 -- good cancellation. So let's say that this was.

00:39:44.450 -- 500KV to 250KV.

00:39:50.810 -- And the cities were.

00:39:53.330 -- 1000 to 5

00:39:56.690 -- in. 500 to 5 so that's something that you could pretty easily.

00:40:00.320 -- Fine cities.

00:40:03.260 -- To cancel that right?

00:40:06.340 -- If we look at 3:45 to 138.

00:40:13.080 -- That's not going to be so easy to find CTS that give

00:40:16.572 -- you a good cancellation on that. So even if this was

00:40:19.773 -- even if these were still.

00:40:22.920 -- Thousands of five.

00:40:27.930 -- This would need to be basically 1000 times.

00:40:33.640 -- 38 / 345.

00:40:37.240 -- To five.

00:40:43.830 -- And chances are that's not going to be a nice stock

00:40:47.108 -- number that you're going to be able to buy in. SNS ET.

00:40:56.510 -- And so it's one that we're we'll talk about a solution for that,

00:41:01.424 -- but this is basically going to

00:41:03.692 -- be. Having

00:41:06.760 -- taps on the relay.

00:41:10.600 -- So watch mechanical relays. What they had was they had multiple

00:41:13.625 -- tap points where you could

00:41:15.000 -- connect. The inputs from the transformer for the differential

00:41:19.010 -- and you could partly correct for that mismatch to a degree you

00:41:23.690 -- couldn't. You could not connect 4 correct for it perfectly, but

00:41:27.980 -- you could. You could go a long ways towards correcting it.

00:41:33.160 -- What we'll see in probably not today. We may. I don't know if

00:41:37.697 -- we get to the example today, what you'll see in

00:41:41.187 -- microprocessor relays now that's just a number, so it's just a

00:41:45.026 -- scaling factor, so you can. So basically you as you enter the

00:41:49.214 -- stuff into the relay for setting it, you're entering the

00:41:52.704 -- information so the relay calculates that tap and you

00:41:55.845 -- don't even have to answer. Calculate it yourself so you say

00:41:59.684 -- OK, here is the MVA rating. Here's the voltage rating.

00:42:03.680 -- And then at the relay says OK and this is the rated

00:42:06.980 -- current and just basically calculates it for you.

00:42:11.830 -- And then you also put the seat. The actual CT ratios 'cause it

00:42:15.444 -- puts that in as a correction to.

00:42:27.670 -- Another thing you'll see in a lot of large power Transformers

00:42:31.080 -- is they have taps, right?

00:42:34.410 -- So we may see.

00:42:37.830 -- 500KV to 250KV.

00:42:42.510 -- Anne, this could be we could

00:42:46.392 -- have. Plus 2 1/2% + 5%

00:42:53.060 --

5%.

00:42:58.620 -- And these could also have some different apps. So if

00:43:01.930 -- you start putting.

00:43:04.060 -- If you and so in some cases, these maybe.

00:43:08.340 -- For lower power ones, these may be on load. Tap

00:43:11.695 -- changing Transformers where they can be changed. In other cases

00:43:14.745 -- the transformer has to be D energized for crew to come in

00:43:18.405 -- and change that tag.

00:43:22.870 -- What is that tap change due to the differential current?

00:43:32.540 -- You just change the ratio of the transformer, right? So you've

00:43:36.984 -- gone to the effort of correcting for compensating for this, this

00:43:41.428 -- ratio and the CT ratios. Now you just threw that off because you

00:43:46.680 -- changed the transfer. The power transformation ratio by 2 1/2%.

00:43:59.070 -- Then another one would be.

00:44:25.660 -- The transformer is always going to draw some magnetizing current

00:44:28.480 -- if it's energized right.

00:44:32.250 -- And this is something that's.

00:44:34.160 -- Going into the transformer and not coming out.

00:44:44.190 -- And as we talked about last time, this might be 2 to 4%,

00:44:48.948 -- maybe 5% of the rated current.

00:45:07.230 -- It will be higher if the transformer is over excited.

00:45:13.210 -- So there's really two things that you need to look at with

00:45:16.414 -- over. Excitation is going to be.

00:45:18.830 -- If the over excitation is severe enough and last long enough you

00:45:23.114 -- want to trip the transformer.

00:45:25.910 -- But you don't want to trip it because you think it's an

00:45:29.414 -- internal fault, so you don't want to trip at the instant it

00:45:32.918 -- happens. So there's some tradeoffs on that, and the

00:45:36.460 -- harmonic content of that's going to be a factor in how

00:45:39.595 -- the relay responds to it.

00:45:44.120 -- Now there's another issue that you have to worry about

00:45:46.480 -- with magnetizing current.

00:45:51.280 -- What would that be?

00:45:58.370 -- So we have magnetizing inrush current.

00:46:09.250 -- So if you energize a transformer.

00:46:23.570 -- You're going to see a current that's going to

00:46:25.568 -- start out looking like this.

00:46:28.260 -- And it may take a second or two

00:46:31.508 -- to. One at one to two seconds to get down to the normal

00:46:36.314 -- magnetizing current.

00:46:40.990 -- So are people familiar? Why Transformers exhibit

00:46:43.867 -- this behavior?

00:46:52.120 -- So it goes down, it goes back to our hysteresis characteristic.

00:46:57.400 -- So the transformer is going to when it's operating is

00:47:00.650 -- going to be.

00:47:03.580 -- Following something that looks like this, right? So if this is

00:47:07.507 -- B versus H.

00:47:10.670 -- This is proportional to voltage. This is proportional to current.

00:47:15.790 -- So every time you go through a sinusoidal cycle, it's going to

00:47:18.982 -- trace this curve, right?

00:47:22.010 -- And so when you deenergize the transformer, you deenergize

00:47:26.042 -- nearer at a current 0, right?

00:47:29.630 -- And so when the current goes to zero, you're going to be

00:47:32.654 -- somewhere up here. And so there's going to be some trapped

00:47:36.706 -- flux on the core.

00:47:38.830 -- When it's deenergized and depending on where you were in

00:47:42.030 -- that hysteresis cycle, when the breaker contact cleared or what

00:47:45.230 -- the power factor of the current

00:47:47.150 -- was. Usually the final invoice and normal routine operation

00:47:51.948 -- when I want to Transformers.

00:47:54.940 -- D energize you open one side, then you open the other ones

00:47:59.476 -- you're interrupting, basically just magnetizing current with

00:48:02.122 -- the final. The energizing of the transformer.

00:48:06.540 -- When you re energize it.

00:48:09.140 -- How is voltage related to flux in a transformer?

00:48:13.830 -- So V is equal to NDF DT, right? So the flux in the voltage or 90

00:48:19.014 -- degrees out of phase with each other. But you can so that the

00:48:23.226 -- voltage here at some point in a sinusoidal voltage waveform you

00:48:26.790 -- can map that the flux when you energize it. So when you're when

00:48:31.002 -- you close a circuit breaker, there's going to be some

00:48:34.242 -- basically effective flux that you're you're trying to impose

00:48:37.158 -- on that core. So if you're lucky and you and you pose a circuit

00:48:41.694 -- breaker in the effective flux for the point on waiver, you're

00:48:45.258 -- closing. It's about what you trapped on the core.

00:48:48.680 -- Then there's not really going to draw any current.

00:48:53.430 -- If you're unlucky and you had trap works up here and you're

00:48:56.562 -- closed when you're somewhere down like this, now the

00:48:58.911 -- transformer is going to draw a lot of current to try to

00:49:02.043 -- equalize that flux. And after magnetizing inrush current.

00:49:06.320 -- And it's very nonlinear current.

00:49:09.000 -- And so this has a lot of harmonic content. The

00:49:12.210 -- generally it's going to be dominated by second and

00:49:15.099 -- then 5th and so on. But it's going to have more

00:49:18.630 -- even harmonics where the over excitation is only

00:49:21.198 -- going to be odd.

00:49:25.610 -- How's the modern steels that they're using in newer

00:49:29.615 -- Transformers? Do not have a sharper second harmonic

00:49:32.839 -- characteristic. They still draw big magnetizing currents, but

00:49:35.135 -- now there's not as clear a second harmonic, and we'll talk

00:49:38.292 -- about some of the issues with that later in the.

00:49:42.650 -- Not this, not later today, but next week or

00:49:45.570 -- the week after next.

00:49:49.040 -- So you've got these very large currents again, they're just

00:49:51.940 -- going into the transformer.

00:49:57.860 -- And so you know, if you're doing

00:49:59.764 -- a normal. Registration of the transformer. Not something

00:50:02.364 -- following like Re closing in a fault. You might have this side

00:50:06.312 -- open and you energize this side and so now you're seeing current

00:50:10.260 -- San people have measured currents as high as 15 per unit.

00:50:16.260 -- If there are a lot of lights, limits that is partly whether

00:50:19.596 -- the surrounding power system can supply that much current.

00:50:22.098 -- If there's too much impedance in the power system that won't

00:50:25.156 -- supply it.

00:50:28.120 -- And so you're doing. You have a differential element. You're

00:50:31.140 -- going to see. Let's say it's something more normal, like 5 to

00:50:34.764 -- 7 per unit for a second.

00:50:37.980 -- So in electromechanical relays.

00:50:41.570 -- One of the things that they did initially was basically turn off

00:50:46.274 -- the differential element until the inrush current period was

00:50:49.802 -- over. They still had issues where if you had two

00:50:53.199 -- Transformers that were close together and you energized one

00:50:55.638 -- when the other one was on, sometimes you had a sympathetic

00:50:58.619 -- trip of the different of the differential element for the one

00:51:01.600 -- that was already energized.

00:51:07.180 -- Professor, I have a question on this one, so

00:51:09.952 -- there is no saturation really, it's just the.

00:51:13.740 -- The core trying to reach that

00:51:15.876 -- flux level. But there's no saturation, so as.

00:51:21.150 -- It face it, it started has sort of a saturation effect because

00:51:24.966 -- of where it pushes the flux, but there really isn't any true

00:51:28.782 -- saturation of the core in this.

00:51:31.560 -- So why isn't it sinusoidal?

00:51:35.990 -- So when you think about the iron in the core right, you

00:51:40.423 -- basically have a bunch of magnetic domains that want to be

00:51:44.174 -- in random directions, right? So let's say that because of the

00:51:47.925 -- trap flux, they're all pointing

00:51:49.630 -- this direction. And for the inrush you're trying to flip

00:51:53.712 -- them all to go back. Basically you want the flux to go this

00:51:58.249 -- way, so you need to flip all

00:52:00.692 -- these domains. And.

00:52:03.920 -- They don't, simply.

00:52:06.420 -- Follow a nice thing in sinusoidal behavior as they flip

00:52:09.250 -- on this. So there's some resistance. I'm really

00:52:12.652 -- oversimplifying this, but basically it's it's a

00:52:15.186 -- magnetic. The nonlinear magnetic behavior of the core

00:52:18.082 -- that keeps it from looking sinusoidal.

00:52:25.980 -- And this harmonic, and So what we're going to see in a little

00:52:30.426 -- bit, is that to try to minimize

00:52:32.820 -- this effect. The second harmonic is often used as a

00:52:37.252 -- as a signature, so if the second harmonics above a

00:52:40.942 -- certain threshold.

00:52:43.030 -- Then it's got the relay will block the differential

00:52:46.189 -- element, so you can either do harmonic blocking or harmonic

00:52:49.699 -- restraint, which is basically making the slope steeper.

00:52:53.590 -- Now, this raises an interesting thing. From a relay point of

00:52:57.572 -- view. We talked about digital filters, right? So here we

00:53:01.192 -- talked about second harmonic. I talked about fifth Harmonic when

00:53:04.812 -- I talked about over excitation detecting over excitation.

00:53:09.520 -- So remember what we talked about with digital filters? If

00:53:12.270 -- we're using cosine filters.

00:53:14.730 -- Well, the is the what is a cosine filter due to harmonics.

00:53:19.866 -- What's the gain about cosine filter 0, right? So the relay

00:53:24.574 -- needs a separate.

00:53:26.820 -- Cosign filter that if you want to measure second harmonic or

00:53:30.582 -- you want to measure 5th harmonic or any of the others, you need

00:53:35.028 -- to have some separate filter elements that are going

00:53:38.448 -- to calculate those.

00:53:40.160 -- Because the normal cosine filter using for your protection

00:53:43.400 -- calculations is going to have a gain of zero and block those.

00:53:49.450 -- And when you start getting up to 5th or 7th, now you're

00:53:52.450 -- starting to get up to the range where the low pass filters,

00:53:55.450 -- anti aliasing filters also going to have an effect on

00:53:57.950 -- them.

00:54:03.460 -- So when you talk about residual magnetism, why doesn't it die

00:54:07.387 -- out? So if I'm.

00:54:09.370 -- I'm switching off or closing opening the breaker in front of

00:54:13.286 -- the transformer at equals to zero. Eventually the residual

00:54:16.490 -- magnetism should die out, right? If I'm not energizing it back in

00:54:20.762 -- let's say days or weeks. So does it die out and not? It does

00:54:25.746 -- decay OK, so basically it's a it's a thermal process. So

00:54:29.662 -- basically these are going to try to randomize if the car is warm

00:54:34.290 -- when you demagnetize it, then they tend to randomize faster

00:54:37.850 -- than if the core is cool as the core as a transformer cools that

00:54:42.834 -- slows down the rate.

00:54:44.460 -- That randomization OK, but even if it's gone to zero an you

00:54:48.900 -- closing your somewhere up here still we're going to have

00:54:52.970 -- some issues on that.

00:54:57.930 -- Awhile back, well actually one of the Masters students here who

00:55:01.989 -- works at Sweitzer. Now guy named Doug Taylor looked at using a DC

00:55:06.786 -- source to preflex the transformer so you could put

00:55:10.476 -- the trap flux at a known at a known point and then if you have

00:55:16.011 -- Breakers with individual phase control then you can control

00:55:19.332 -- when you close them.

00:55:22.220 -- They also are using variations of that an like.

00:55:28.760 -- There's been a lot of stuff looking at that in Europe, for

00:55:32.324 -- example, in some of the offshore wind farms where they basically

00:55:35.591 -- are in a system that can't supply that magnetizing current

00:55:38.561 -- to magnetize the core, because there isn't a source strong

00:55:41.531 -- enough to provide it out there.

00:55:44.090 -- And so they want to be able to close the Transformers with no

00:55:48.276 -- inrush. And so rather than pre flexing the cores, they're

00:55:52.692 -- looking at trying trying to dissipate the flux in the

00:55:56.960 -- core so that they can bring it to zero, and then they do

00:56:02.004 -- individual phase control on the Breakers to minimize the inrush.

00:56:07.670 -- Also the whole pre fluxing minimize trying to get the

00:56:10.730 -- known side of inrush makes a big difference. If you have a

00:56:14.402 -- five legged core versus the three legged core.

00:56:18.190 -- So when you see the anticipated, basically they figure out at

00:56:21.644 -- what time or what voltage at what point in the voltage the

00:56:25.412 -- breaker was opened, and then based on that they calculate the

00:56:28.866 -- residual magnetism and the decay, and then they open

00:56:31.692 -- individual phases at different times. Or they close them, they

00:56:34.832 -- close them at specific times. OK, so the Breakers are always

00:56:38.286 -- going to try to open it. A natural current 0. Sure, an

00:56:42.054 -- there are actually some big problems if you don't open it in

00:56:45.822 -- natural current 0, because then you can get very big.

00:56:49.350 -- Transient response if you do a current shopping.

00:56:53.590 -- So the parasitic capacitance of the winding will interact

00:56:56.560 -- with the magnetizing branch, and you can see like 2 / 2

00:57:00.520 -- per unit voltage.

00:57:03.600 -- Even if you're chopped like half an amp.

00:57:11.700 -- That's a topic more for you. See 524 though.

00:57:20.290 -- OK, so any other questions related to the magnetizing.

00:57:24.950 -- Current behavior.

00:57:27.630 -- So these are all things that need to be accounted for in

00:57:31.758 -- creating the differential element an in setting like

00:57:34.510 -- the slope and the minimum operate current.

00:57:39.090 -- The other one to look at is going to be the transformer

00:57:42.342 -- phase shift.

00:57:49.760 -- So I started out drawing a YY transformer.

00:58:00.000 -- So the other thing we have to look at is Delta Y.

00:58:04.150 -- Or why Delta Transformers?

00:58:22.310 -- And so in North America there's an ANSI IEEE standard so that

00:58:27.926 -- the phase shift is generally very predictable, right?

00:58:33.330 -- And what's the standard?

00:58:37.720 -- Sorry. The high side is leading by $30.

00:58:59.020 -- So V line the neutral in the high voltage side leads

00:59:01.902 -- vilanda neutral in the low voltage side by 30 degrees.

00:59:06.370 -- The Power systems textbook I used when I was an undergrad

00:59:10.055 -- gave the impression that whenever you had a Y Delta

00:59:13.405 -- transformer or the Y side always led the Delta side by 30 degrees

00:59:17.760 -- because the author in.

00:59:20.620 -- All the cases he had run across the Y side was always

00:59:24.328 -- a high voltage transformer, 'cause he'd always worked in

00:59:27.109 -- transmission and never worked in distribution.

00:59:38.430 -- And so. So one of the effects were going to have

00:59:42.274 -- obviously is the 30 degree phase shift this also.

00:59:58.400 -- The Delta Y connection also

00:59:59.820 -- impacts the. Turns ratios right. So now you've got this other

01:00:03.574 -- sqrt 3 that gets put in there in addition to having.

01:00:11.110 -- The voltage transformation ratio.

01:00:14.910 -- That sqrt 3 shows up in the current so that reflects

01:00:18.320 -- back to the CTS.

01:00:23.620 -- And let's say that we have a Delta Y grounded transformer.

01:00:28.640 -- So this side.

01:00:41.830 -- When we're measuring the phase currents, there's going to be 0

01:00:45.537 -- sequence current on this side, but there won't be on this side.

01:00:53.200 -- And so even some Even so, one of the things that you have to be

01:00:57.550 -- careful of his solutions to try to fix this phase shift.

01:01:01.490 -- And fix this also after account for this. So I said that they

01:01:05.871 -- are one of the solutions that people did for less mechanical

01:01:09.578 -- relays. Had to have an extra step added to it because of

01:01:14.094 -- the zero sequence kind.

01:01:26.250 -- So if we have a transformer.

01:01:47.130 -- So we can look at the CTS.

01:01:51.140 -- So for electromechanical relays.

01:02:00.880 -- The common solution in this for this was going to.

01:02:06.860 -- To use the CT connections to help cancel for the cancel this.

01:02:12.520 -- And so.

01:02:16.250 -- So one option.

01:02:31.830 -- Would be to connect the CTS on the Y grounded side in Delta.

01:02:38.340 -- And the CTS and the Delta side and Y.

01:02:57.440 -- You need to make sure you connect the Delta properly to

01:03:01.092 -- cancel the shift. But So what that means is that the that the.

01:03:07.110 -- Phase currents that the Delta phase currents.

01:03:12.580 -- Well, include the zero sequence current that's going

01:03:15.148 -- to circulate in that Delta, but then the line currents

01:03:18.358 -- coming off the Delta which go to the differential relay

01:03:21.568 -- will not have.

01:03:23.640 -- That current

01:03:29.600 -- morning your device is running low on memory.

01:03:37.470 -- So one of my colleagues has a sledgehammer. He brings the

01:03:40.737 -- class for people whose cell phones make noise during class.

01:03:47.100 -- The new phone is trying to shut it down.

01:03:52.290 -- And so this is so, you still will run across substations that

01:03:57.018 -- have the CTS wired this way from the electromechanical relays.

01:04:03.920 -- And then a second option.

01:04:16.440 -- Would be the connect.

01:04:18.660 -- This it is an Y and this it isn't Delta.

01:04:24.240 -- So yes, there's a problem with this one, right?

01:04:30.810 -- So now the.

01:04:35.270 -- The differential element on this, the current that goes to

01:04:38.140 -- the differential an element from this side, it's going to include

01:04:41.297 -- zero sequence current. The one in this one won't, right.

01:04:45.970 -- So this one is going to need.

01:04:55.020 -- So basically this one needed an auxiliary set of current

01:04:58.210 -- Transformers that would block the zero sequence current by

01:05:01.081 -- basically circulating it in the auxiliary Transformers and

01:05:03.633 -- not have a go to the differential element.

01:05:26.570 -- So now if you go to a substation where it's new

01:05:31.553 -- construction and it's designed not anticipating

01:05:34.271 -- that there's going to be microprocessor relays

01:05:37.442 -- protecting this.

01:05:47.680 -- Now the seats are going to be why on both sides and there

01:05:51.632 -- will be a ground reference in the seat path.

01:06:19.860 -- And it will also the CTA will basically perform calculations.

01:06:24.340 -- To compensate for the phase shift an it's going to

01:06:28.920 -- perform another calculation to remove I 0.

01:06:35.480 -- And these are actually going to be matrix multiplications.

01:06:48.380 -- So I have a handout that.

01:06:51.350 -- Maybe I will pass it out today. You need to

01:06:53.950 -- remember to bring it.

01:06:57.250 -- Don't be sorry.

01:07:13.600 -- And so.

01:07:18.360 -- This first calculation is basically.

01:07:23.660 -- Typical calculation that you would see.

01:07:27.220 -- Done in the relay.

01:07:29.840 -- For the.

01:07:32.120 -- As an intermediate step for going to the

01:07:35.264 -- differential element.

01:07:37.540 -- So you're gonna have.

01:07:40.570 -- You're going to have the primary currents. Then they're going to

01:07:44.112 -- be divided by the current

01:07:45.722 -- transform transformation ratio. Remember, these are

01:07:48.686 -- why connected.

01:07:54.000 -- And then there's also going to be this tap calculation, and

01:07:57.744 -- the other hand out goes into more detail about the how this

01:08:01.488 -- tap is calculated. And then there's going to be a correction

01:08:06.020 -- matrix, so the correction matrix the output is going to be the

01:08:09.920 -- secondary current with the phase and zero sequence correction.

01:08:16.110 -- And so the current from both windings are going to. So this

01:08:20.190 -- is actually. This would be the primary side, and then we're

01:08:23.930 -- going to secondary sidewinding. So this is actually.

01:08:27.470 -- The power transformer primary.

01:08:53.070 -- And then the correction matrix, or a number of correction matrix

01:08:58.240 -- we can do. And so when I say matrix zero, that is using the

01:09:04.820 -- IC Clock terminology. So if we think about o'clock, we're going

01:09:09.990 -- to have 12369, etc and then 12.

01:09:13.890 -- 12 is also equal to 0, right?

01:09:19.820 -- And so if we have a Y connection with, if you say

01:09:24.212 -- that we have basically our phase, a voltage is going to

01:09:28.238 -- be here at an angle of 90 degrees. That's our zero

01:09:32.264 -- position.

01:09:37.340 -- And so the Matrix Zero is assuming we have a Y

01:09:40.783 -- connection and we're not trying to do any reversal of

01:09:43.913 -- the voltages, so this will be just the identity matrix.

01:09:53.370 -- And then where matrix one is the one o'clock position and this is

01:09:59.129 -- one that in.

01:10:00.540 -- South America is often referred to as the DAB and

01:10:03.490 -- this would be a Delta.

01:10:08.250 -- AV connection so that means that the first winding of the

01:10:11.583 -- Delta is connected from A to B. The second line will be to

01:10:15.522 -- see the third one will be see to a. This gives you remember

01:10:19.461 -- North America. You're limited to either plus 30 degrees or

01:10:22.491 -- minus 30 degrees when you're going from Y to Delta. So all

01:10:26.127 -- we care about in North America is going to be the D1

01:10:29.763 -- in the D11 connection.

01:10:33.240 -- And then we have the D11 connection, and so if we

01:10:37.398 -- compare these all it's doing is exchanging

01:10:40.044 -- which rows are have the.

01:10:43.410 -- Then have the different column combinations.

01:10:47.840 -- And so, well, we'll talk about this a little bit more, applying

01:10:52.172 -- it in the other example.

01:10:54.970 -- And then, as I mentioned, we have that we need that zero

01:10:58.054 -- sequence removal matrix too.

01:11:03.900 -- And so that's what this one does.

01:11:08.460 -- And so this is mathematically reproducing

01:11:10.410 -- the effect of the current circulating in the Delta.

01:11:20.750 -- Anworth this what this is coming from?

01:11:24.700 -- A very good reference for summarizing this is.

01:11:31.420 -- A paper that was written by.

01:11:35.230 -- I group from Basler Electric John Horack.

01:11:37.659 -- Actually, I have a link to on their class links web

01:11:41.476 -- page. I have a link to webpage it he's got put

01:11:45.293 -- together an extensive web page was protective

01:11:47.722 -- relaying. Related links.

01:11:51.260 -- And so I did not. I gave you copy. It's, uh, some of the

01:11:54.676 -- pages from this paper. I have links to the whole paper on

01:11:57.604 -- the course web page. That's the on campus students. There

01:12:00.044 -- were some of the pages that I'm going to talk to talk

01:12:02.972 -- about today and next time.

01:12:06.810 -- So this is just showing sort of the connection information

01:12:10.070 -- as a reference for the rest of this paper.

01:12:16.410 -- So.

01:12:18.450 -- He has uppercase letters to indicate the primary lowercase

01:12:22.635 -- to do the secondary.

01:12:25.520 -- And then he has the third of the terminal ends an the.

01:12:31.060 -- So this would be the polarity end of the wine,

01:12:33.656 -- and this is the nonpolarity end of the winding.

01:12:40.150 -- And so. This is one of the things that you go through.

01:12:45.030 -- You're going to find different people in different places, use

01:12:48.580 -- somewhat different notation so we see UV WABC.

01:12:52.070 -- And so on.

01:12:59.290 -- And so if we wanted to build a YY transformer in a typical

01:13:05.166 -- North American connection so when we see the W1W 2W3, those

01:13:10.138 -- are referring to the winding.

01:13:14.330 -- The windings of the six windings that produced the

01:13:17.372 -- three phase transformer.

01:13:21.590 -- And then it's not very obvious, but these are his

01:13:24.910 -- polarity marks for those windings.

01:13:28.910 -- And so H1X1 this is high voltage. This is

01:13:31.664 -- low voltage and so on.

01:13:34.330 -- And so mapping these this is how they would map.

01:13:39.870 -- Tell the two winding sets.

01:13:47.510 -- And so winding one and winding 4 on the same course.

01:13:50.282 -- So these two are going to be in phase with each other.

01:13:55.700 -- And so you can use this to build the diagram for how

01:13:59.168 -- the transformer ones relate to how the windings relate

01:14:01.769 -- to each other.

01:14:06.900 -- And so then he goes on to look at.

01:14:15.830 -- So the basically the Y zero is the one that's most

01:14:19.669 -- common in North America.

01:14:24.100 -- And so we can look at things that change polarities by so

01:14:27.556 -- the Y four is now we're shifting things down to the

01:14:30.724 -- 4:00 o'clock by putting winding one connected to Phase

01:14:33.316 -- V.

01:14:35.850 -- White and then we can just look at all these different

01:14:39.546 -- combinations. WHI Six is just reversing the polarity so the

01:14:42.906 -- polarity marks reversed unwinding one.

01:14:47.440 -- And so this is another one that is more of an industrial

01:14:51.076 -- power systems one, but you'll sometimes see Transformers

01:14:53.500 -- with wired opposite of the polarity marks.

01:14:57.580 -- Then he goes through the same thing with Delta windings.

01:15:02.450 -- So the. And so next time we'll go back and look at

01:15:06.270 -- this in terms of a Y Delta transformer. How we do the

01:15:09.054 -- plus 30 if the Y is a high side, how we do the minus 30?

01:15:12.534 -- If the why is the low side?

01:15:17.010 -- And so this paper goes on to kind of lead into deriving

01:15:21.402 -- those connection matrices.

01:15:24.560 -- And so we'll finish talking about this paper next time, and

01:15:28.014 -- then we'll talk about the.

01:15:31.130 -- Example handout so that we're going to apply these

01:15:34.622 -- connection matrices to measurements for a fault.

01:15:38.450 -- We can look at an internal fault or an external fault. We

01:15:42.458 -- can also look at what happens if somebody accidentally left

01:15:45.798 -- ascete shorted in the substation and how that plays

01:15:48.804 -- through these connection matrices.

01:15:51.560 -- So with that, well, any questions before we stop.

01:15:55.730 -- OK, and just a reminder for the outreach students.

01:15:58.115 -- There is no class on campus next week, so there will be

01:16:01.295 -- no new lectures for a week.

01:16:05.650 -- OK, that's all done.

0:16
Hi everyone. Welcome to EM 513. In our session today, we're going to talk about great by choice by Jim Collins and it really is a good follow on to the last session we had where we talked about.
0:33
Way by choice, and hopefully you're going to see some similarities of the concepts. But there's also a little different twist that I'd say comes out of the research that was done for this book. And hopefully, though, you'll see how those things really do hang together pretty well. And.
1:00
Seemed to make a lot of sense.
1:04
So let's.
1:09
Here we are again in our overall course framework. We are in this last session on leading organizations. After today, we'll have two more sessions that will where we will focus on the concept of.
1:28
Execution and really applying good discipline to getting things done within the broader organization, which of course will be an extension to a lot of the concepts we talked about in leading others. And then the final session of the class will be focused on building the leadership pipeline within your business. We'll talk at that point about transitions you may be going through.
1:57
As, as even you yourself move from.
2:01
Or move along a management career path or leadership career path and we'll talk about how you can do the right things to invest in your team and really make sure that you have a leadership pipeline that's going to set you up for.
2:19
Having good successors, and we talked about that in the last session. And if you do have the chance to read rate by choice, there's some great stories in there about how people did in fact set themselves up for.
2:35
Some good.
2:37
Succession planning, if you will.
2:42
I wanted to at least start with, as a refresher, the good to great framework that we talked about last time and I'll I'll try to make references here.
2:53
Where I think it's appropriate, and if you recall the high level framework that came out of the research in good to great was this notion of having disciplined people, disciplined thought and disciplined action. And we talked about 2 concepts within each of those areas and hopefully you'll see the connections as the work is extended to the research done for great by choice.
3:24
Here is a little bit of background on the research that.
3:29
Took place for great by choice. It was a 9 year research project.
3:34
That was started in 2002, and it was based on the question why do some companies thrive in uncertainty, even chaos, but others do not?
3:50
The high performing study cases that were examined in the book were called 10 Xers.
3:59
The research done for built to last you know if you recall really looked at companies that had been successful over this 15 year time frame.
4:12
In good to grade, the research was similar, but it also looked at another dimension. It looked at the extremity of change that was happening in the particular environment that these businesses were in and that was a critical element that was explored along with the performance of each of these companies.
4:39
What is A10 xer in the context of this book?
4:43
It is a company that has sustained what Conn's calls truly spectacular results for an era of 15 plus years relative to the general stock market and also relative to its own industry.
4:58
The enterprise achieved these results in a particularly turbulent environment.
5:04
Full of events that were uncontrollable, fast moving, uncertain and oftentimes harmful to the businesses.
5:14
The companies studied began their rise to greatness from a position of what they describe as vulnerability.
5:23
Where they were either young or a pretty small company at the start of this journey.
5:32
They started by examining over 20,000 companies. They had a set of criteria and ultimately cuts that they used to continue to whittle down the list. In fact, they went through 11 layers of cuts to get to this final set of 10 experts.
5:55
It didn't include the list here because my assumption is you're all going to.
6:02
Read this book.
6:05
On Page Six of the book identifies the final set of business cases, and quickly here I'll read these Amgen, Biomet, Intel, Microsoft, Progressive Insurance, Southwest Airlines, and striker.
6:25
Many of these of course are very familiar to us and they certainly are in businesses that we would say are either.
6:35
Very competitive. I'm tech. Lots of competition.
6:40
Significant changes, etc.
6:43
And that those things all contributed to their ending up on this final set of TENEX cases.
6:52
Interestingly enough, the TENEX cases during the comparison time frame.
7:00
Or during the timeframe assessed and with the comparison companies, they performed 30 times better.
7:09
Which is really pretty phenomenal.
7:14
The framework that Collins is going to use as a result of the research done here is really this triangle I have in the center of the slide I put in just as a starting point again the framework that we saw in good to great and.
7:32
The three key areas there were disciplined people, disciplined thought, discipline, action.
7:37
And.
7:39
Level 5 leadership of course was a critical element discussed as a part of disciplined people and you can see they carry forward this level 5 leadership. They are calling it level 5 ambition in this case and what they really highlight the most or maybe in addition to the characteristics we saw discussed last session are this idea.
8:07
Or is this idea of inspired motivation?
8:11
And.
8:13
In a particularly turbulent environment where there's a lot of change happening.
8:19
The motivation.

Duration:"00:52:28.7600000"



00:00:30.200 -- Yes.

00:00:33.030 -- So today we will continue discussion about the

00:00:35.870 -- modified, all the method and Runge Kutta methods. So we

00:00:39.420 -- will talk about the formulas and then accuracy and so on.

00:00:43.325 -- So I give you hand out and the problem. I'll use it

00:00:47.585 -- today so that we can cover a little bit faster. And then

00:00:51.845 -- I'll spend time on other material. OK, so.

00:00:58.460 -- In there you remember in all this method in order to go from

00:01:03.751 -- point X&YN to point XN plus one 1 + 1, essentially with another

00:01:09.042 -- next index, we only use information from the previous

00:01:12.705 -- point. So in a modified or leave use information from 2 points

00:01:17.589 -- and we use oil as step to go to the point X N + 1 NU N +

00:01:24.915 -- 1. This is predicted point.

00:01:27.950 -- And then be available slope at the predicted point and we use a

00:01:33.059 -- slope at initial, not initial. But the point that we start

00:01:37.382 -- start from and then we average these slopes defined slope

00:01:41.312 -- alone, which we find essentially construct line right tangent

00:01:44.849 -- line and then we find approximation at the next step.

00:01:48.779 -- So I also wrote this method last

00:01:51.530 -- time. So you can either define predictor which is the Oilers

00:01:57.000 -- step and then this is slope at.

00:02:01.370 -- .1 right and here we have slope at .2 and then we average slopes

00:02:07.082 -- and this is how we find the next. The next point all we can

00:02:12.794 -- write down these slopes explicitly. So K1 is a slope at

00:02:17.282 -- point. XNYN and then we use it to March to find point you and

00:02:22.907 -- plus one. Then we find K2 slope at the second point and then we

00:02:27.793 -- take every to the slopes defined. And if you don't want

00:02:31.632 -- to use K1K2 and just write this in terms of an even without you

00:02:36.518 -- N + 1, then you just write explicitly all the expressions

00:02:40.357 -- for you and plus one. So this is a first step. Predictor

00:02:44.894 -- does not change and in the second step in the character.

00:02:48.970 -- You have your own plus one equals UN plus H / 2, so you

00:02:53.702 -- take average. This is your slope at point XYN. This is your point

00:02:58.096 -- and you X N + 1 right here. This is your predicted point. U N + 1

00:03:03.842 -- essentially just written

00:03:04.856 -- explicitly. OK.

00:03:09.510 -- Modified the oldest method uses two term approximation from the

00:03:13.990 -- Taylor series right. The constant term and the linear

00:03:18.022 -- term. The modified Euler method uses. Also next terms uses

00:03:22.502 -- quadratic term in the Taylor expansion, so if we go back to

00:03:27.878 -- their tail expansion then modified Euler will use up to

00:03:32.358 -- age squared term. So this means that the first time that you

00:03:37.734 -- neglect will be proportional to

00:03:39.974 -- H cube. Right next will be age to the 4th. Each of the 5th and

00:03:45.450 -- if H is small then this will be a dominant term. So air local

00:03:50.070 -- error over one step will be proportional to H cube.

00:03:54.290 -- And then you find cumulative error after multiple steps right

00:03:58.000 -- after. If you're going from zero to X final, then the error will

00:04:02.823 -- be proportional to age squared, so similar usually you lose one

00:04:06.904 -- order when you sum the errors you find cumulative error. So

00:04:10.985 -- since modified term all this but it matches the 1st three terms

00:04:15.437 -- in the Taylor series up to and including termination squared,

00:04:19.147 -- the local area is proportional to each cube, but the cumulative

00:04:23.228 -- error is proportional to age

00:04:25.083 -- squared. So if air is proportional to H squared

00:04:28.998 -- and instead of H, you take H / 2, what would happen

00:04:32.910 -- with the error?

00:04:35.610 -- As will decrease by approximately 1 force, right? So

00:04:38.265 -- if you see so, this is a way how you can check that your method

00:04:42.690 -- is quadratic. So this means that your method is quadratic,

00:04:45.935 -- so your error is proportional to each squared. Let's say you

00:04:49.180 -- write a program and how would you verify that? Yes, the method

00:04:52.720 -- is programmed correctly. So what you can do you take you take a

00:04:56.555 -- test problem for which you know exact solution, so you can look

00:05:00.095 -- at the error because error would be the difference between exact

00:05:03.340 -- solution and numerical solution. So you go from.

00:05:06.430 -- Initial time to some final time final point, and you compute

00:05:11.457 -- solution at the final point.

00:05:14.530 -- And you look at the error right? And then you decrease error by

00:05:18.391 -- half and look how the error will change. So if error bill

00:05:21.955 -- decreased by half, this means that you have a linear method.

00:05:26.250 -- If it decreased by quarter, than its accuracy is quadratic.

00:05:32.310 -- OK. So this is a way to verify that your program is correct,

00:05:37.448 -- and then once you verify your code then you can change

00:05:41.034 -- equation. You can change function, then you can more or

00:05:44.294 -- less thing that your program is reliable, computes correctly, so

00:05:47.554 -- this is what happens with the error. Is H decreased by half

00:05:51.466 -- then the arrabelle because by a factor of four and just for

00:05:55.704 -- comparison. Again for all this method it's a linear

00:05:58.638 -- convergence. So if you decrease age by half your error will also

00:06:02.550 -- decrease approximately by half.

00:06:05.710 -- OK.

00:06:11.410 -- And so essentially we know the methods we just. I can just

00:06:15.442 -- rewrite it may be in the way that is more convenient for

00:06:19.810 -- programming. So if we want to solve initial value problem with

00:06:23.506 -- some initial condition. So what do we need? We need initial

00:06:27.202 -- condition right? So X not, why not? We also know we need to

00:06:31.570 -- know the step size and how many steps we have to perform right

00:06:35.938 -- function F is known. So once you have equation you can find

00:06:39.970 -- function F so again.

00:06:41.420 -- Then before you need to compute 'cause you have some homework

00:06:46.051 -- that you have to actually implement by hand or using

00:06:50.261 -- Calculator. So write down the formulas before you substitute

00:06:54.050 -- values right so?

00:06:56.320 -- You can, we can use either write this in terms of predictor

00:07:00.256 -- corrector or we can use this slopes K1K2 to write the method

00:07:04.192 -- so XN plus 1 = X N plus H. So every time you increment by H

00:07:09.440 -- right and also we can write

00:07:11.408 -- that. H is X final minus X starting divided by number of

00:07:17.326 -- steps right or number of steps is X final minus 0 / H right?

00:07:23.150 -- So if you know number of steps you know initial point

00:07:28.142 -- terminal point then you can find step size or vice versa.

00:07:32.718 -- If you know step size you can find number of steps.

00:07:39.730 -- OK, predict this step is just the oldest method.

00:07:43.490 -- Right and then corrector? So predicted allows you to find

00:07:46.940 -- this predictive point you N + 1 and then corrector will find

00:07:51.080 -- slopes at both points and average them to find exponent.

00:07:55.210 -- OK, and again Alternatively this is using the K1K2 and but

00:07:59.687 -- essentially the same.

00:08:01.550 -- OK, so whatever way you prefer, you can use.

00:08:08.170 -- OK, any questions here.

00:08:14.070 -- So let's look at the example.

00:08:16.890 -- So in this example you have to implement modified order in.

00:08:23.520 -- And solve the problem in 2

00:08:24.972 -- steps. So equation is Y prime equals X + y -- 1 squared.

00:08:30.250 -- Initial condition by 0 = 2. So find Y at. So you start from X

00:08:35.575 -- equals. O you go to X = 0.2 in two steps means that step

00:08:42.728 -- step sizes. 0.1 right again, it's a 0.2, so H is 0.2

00:08:49.380 -- -- 0 / / 2 zero point 1 which is written here.

00:08:56.710 -- Initial condition X00Y0 stole from here number of steps two

00:09:02.050 -- and then H you find.

00:09:06.260 -- Their function function F function F is the right inside

00:09:09.760 -- of your equation.

00:09:12.550 -- OK, and I know it's tempting to write down right away their

00:09:17.230 -- solutions, but take some time. Just write down the formulas in

00:09:21.520 -- terms of X&YN, it's easier than to substitute. I mean, if you

00:09:26.200 -- program something then you just program with indices and then it

00:09:30.490 -- computable repeat, write your computations. But when you do by

00:09:34.390 -- hand then you have to keep track of X0X1Y0Y1 and then here you

00:09:39.460 -- have you also UN to worry about.

00:09:43.700 -- So you write down the formula. So this is your next.

00:09:46.850 -- Approximation of X. This is your predicted value just

00:09:50.540 -- using the Euler's method, because this is your function

00:09:54.230 -- F at X&YN and then.

00:09:57.420 -- This is your next approximation.

00:10:00.110 -- By using the previous and the average of slopes.

00:10:04.030 -- OK.

00:10:06.520 -- So for all this method to go from one point to another, you

00:10:11.109 -- do one is 1 stage method because you only use one point for the

00:10:16.051 -- modified order, it is 2 stage because you have predictor an

00:10:19.934 -- you have character. So each step has two parts.

00:10:24.160 -- OK.

00:10:26.920 -- So if we take so here, we have N equals.

00:10:33.010 -- Zero, so when N = 0, I have X 1 = X O plus H. We find 0.1,

00:10:40.462 -- which is what supposed to be predicted point Yuan Yuan plus

00:10:45.016 -- one. Will you one and then it's Y0 plus HX0Y0 and you substitute

00:10:50.398 -- values you get 2.1. So this is your predicted value and then

00:10:55.366 -- you can use it in the next stage

00:10:58.678 -- defined. Correction, OK, so this is your essentially. This is the

00:11:02.650 -- same as what you have here.

00:11:05.450 -- So it might be more beneficial to use key one key two if you

00:11:10.014 -- want to reduce time on writing because you have to rewrite

00:11:13.600 -- this. And this is your slope at the predicted point. Again, just

00:11:17.512 -- write down X0Y0X1U one before you substitute values, because I

00:11:20.772 -- mean you see that becomes messy.

00:11:29.940 -- OK, so then we substitute values and we obtain approximation. So

00:11:33.845 -- so we did two stages, but this is the first step.

00:11:38.670 -- OK, it's not 2 steps first step. So now we use N = 1 and

00:11:45.525 -- this allows us to find X2U2 and Y2. So X2 is exam plus H, so

00:11:52.380 -- we have you too is a prediction using the Oilers step from Point

00:11:58.321 -- X one U-1 and then you do is correction with average of

00:12:03.805 -- slopes. Again as you see, right down X one U1X1X2U2 and so on.

00:12:09.880 -- And then approximate and then substitute values.

00:12:16.130 -- So finally so this is our approximation of a solution at

00:12:19.639 -- 0.2, and again this is not exact value, right? It's only

00:12:23.148 -- approximation because we use out of infinitely many terms in the

00:12:26.657 -- Taylor series, we only use 3.

00:12:29.290 -- So H is finite, right? So definitely we have an error. OK,

00:12:33.442 -- so schematically what is going on here? You start. Your initial

00:12:37.248 -- condition was at 02 right? This is your point.

00:12:42.180 -- Predictor brings you to point X one U-1.

00:12:47.560 -- You find slope at this point at X1. You want you find slope at

00:12:54.224 -- X0Y0. You average corrector gives you point X1Y1.

00:12:59.420 -- This is your first step, but

00:13:01.412 -- still stages. Then again from point X1 U one you find

00:13:06.648 -- predictor X2U2 right YouTube means has index as Y two. So

00:13:11.510 -- please different letter. But it is the same index and then

00:13:16.372 -- you've added slopes at X 11X2U2 average them and this

00:13:20.792 -- gives you correct correction point X2Y two again two stage

00:13:25.212 -- but it's one step.

00:13:31.180 -- OK.

00:13:33.760 -- Any questions here?

00:13:36.930 -- So example have either Euler or modified Euler method to

00:13:40.870 -- implement by hand, which means the step size will be generously

00:13:45.204 -- large, maybe like one or something that doesn't require

00:13:48.750 -- because you cannot use calculators for the test there

00:13:52.296 -- 'cause I don't know which device you bring mini. Something

00:13:56.236 -- computer that has access online and so on. So the algebra will

00:14:00.964 -- be simple enough that you can do

00:14:03.722 -- by hand. But for me, even if you have to perform 2 steps.

00:14:08.650 -- I need to see that yes, you know what is initial

00:14:11.411 -- condition. What is the next point and so on. So it will

00:14:14.423 -- not be a lot of steps, but at most Euler or modified Euler.

00:14:18.810 -- OK, your homework has more steps to perform, so you're welcome to

00:14:23.358 -- use whatever calculators computers to get the values, but

00:14:26.769 -- you have to write down. Then you can probably minimize number of

00:14:31.317 -- things that you write.

00:14:33.600 -- OK, your project Modeler project is based on

00:14:36.888 -- implementing these methods actually not implementing.

00:14:39.354 -- Using them to solve problems because the

00:14:42.231 -- programs functions are available on the course

00:14:45.108 -- websites. You just have to.

00:14:49.070 -- Maybe on Monday I'll bring the laptop so I'll show you where

00:14:52.646 -- files are and how to use them.

00:14:57.500 -- OK so next method.

00:15:00.250 -- To consider is so called 1st order on the quota method.

00:15:06.320 -- And the idea here is the falling. So we saw from their

00:15:10.880 -- modified all the method that if we use information from two

00:15:15.060 -- points then we get more accurate

00:15:17.340 -- approximation. Right, so can we use more points to get the even

00:15:22.577 -- more accuracy and the question the answer is yes. So in this

00:15:27.101 -- case we use four points.

00:15:29.560 -- So we go from .1.

00:15:32.980 -- 2.2 Essentially this is your order step. We get point .2.

00:15:38.645 -- Then we use this slope K2 to go to .3.

00:15:44.670 -- We use the .3 slope. Do you go to .4 and then we take weighted

00:15:50.790 -- average of the slopes at this

00:15:53.238 -- point? OK, so OK.

00:15:57.250 -- Um?

00:16:00.400 -- So which points we use? We use

00:16:03.529 -- point X. We use point in the middle of this interval at X N +

00:16:08.925 -- H / 2 and here we have two points to use and we also use

00:16:13.050 -- point at X = N + 1.

00:16:16.350 -- So do we? Do we give the same weight essentially the sum of

00:16:21.316 -- slopes over 4? No, we give twice more weight at points

00:16:25.518 -- in the middle.

00:16:36.200 -- And this is last page that you have an I I did not print. I

00:16:42.095 -- have a few more pages, but.

00:16:45.910 -- I'll explain what we have here. So if you have.

00:16:51.170 -- Probably let me use, maybe this so you don't

00:16:55.094 -- have this page, but this is a recap of the last page, so you

00:16:59.672 -- have to want to solve the 1st order equation with some given

00:17:03.596 -- initial. So I'll bring a copy of

00:17:05.885 -- this next time. So what you do you find the slope at .1. This

00:17:11.178 -- is where you start.

00:17:13.540 -- Then you match half step to .2 using this slope.

00:17:19.110 -- So you have you have X N + H over to you. This is your X

00:17:25.462 -- displacement an in. Why you do Oilless step with step size H of

00:17:30.623 -- it but slow K1.

00:17:33.320 -- So once you have this point, you use this point

00:17:37.380 -- to evaluate slope.

00:17:39.970 -- So I compute slope K2 and I find

00:17:43.858 -- .3. By marching again from KXAN half step and using alone Def

00:17:50.700 -- line with slope Cato.

00:17:53.650 -- OK, this gives me point X 3.3, so from .3 then we match full

00:18:00.090 -- step to find point for using Slope case 3.

00:18:05.020 -- Once you have all these four slopes, you have weighted

00:18:08.630 -- average so you have you give weight 1 to the first point and

00:18:13.323 -- to the last point, but two weights to the .3 and two and

00:18:18.016 -- three. So overall you have for slopes six slopes. So you divide

00:18:22.348 -- age by 6.

00:18:24.130 -- So this is your average weighted slope.

00:18:28.090 -- OK, and then you can write this slope like even if you don't

00:18:32.640 -- know this. So you use information from four points. OK

00:18:36.140 -- to find, so this is a full stage

00:18:38.940 -- method. Anne.

00:18:43.250 -- In order to go from X&YN 2 X N + 1 one plus one, it is still

00:18:49.098 -- using only one previous point, right essentially, but it does

00:18:52.538 -- it in four in four stages.

00:18:55.260 -- OK.

00:19:01.760 -- OK, So what I can say here is there wrong accoutre

00:19:08.756 -- force order matches there?

00:19:14.230 -- The local error in their own decoder 1st order method is of

00:19:18.406 -- order H as a power 5.

00:19:21.980 -- OK, but when you find cumulative error then you lose one order

00:19:27.476 -- and then overall the error is.

00:19:31.300 -- Proportional to H is about four and you can. You can appreciate

00:19:35.116 -- it if H is let's say 0.01 to 10 to the point is the power of

00:19:40.204 -- negative one right? All this method will have error also of

00:19:43.702 -- the order of 10 to the minus

00:19:45.928 -- one. Right modified order will have error to the order 10 to

00:19:50.999 -- the minus. Two but longer code will have error of the order 10

00:19:56.205 -- to the minus four right? So you see that it's occasionally.

00:20:00.180 -- Logic difference in the in the accuracy. So all this method in

00:20:03.744 -- order to get the same accuracy.

00:20:06.480 -- You need to use smaller H. Ruby code allows you to use larger

00:20:11.992 -- step size. Because the error is small and So what you save,

00:20:17.542 -- you save the number of steps. But again, remember that one

00:20:21.634 -- step of the longer quota has

00:20:23.866 -- four stages. So at each stage you have to evaluate function

00:20:28.565 -- and function evaluation may be consuming, so that's so. That's

00:20:32.015 -- why it's not very cheap method because at every step you have

00:20:36.155 -- four function evaluations.

00:20:39.020 -- OK.

00:20:40.850 -- How do we check that method is first order accurate? If we

00:20:45.686 -- decrease H by half, their level decreased by a factor of.

00:20:55.410 -- If H is replaced with H / 2, so the arrabelle decreased

00:20:59.622 -- by a factor of.

00:21:03.430 -- 22 to the power. 416 right so this is, you see, is a

00:21:08.929 -- significant difference between this method and that method OK?

00:21:14.650 -- Which method you would like to use if you have

00:21:17.570 -- to solve your problem?

00:21:22.740 -- So you have a choice. You have three methods and you have to

00:21:27.095 -- implement MCF thread programs, foiler for modified or Lefranc

00:21:30.110 -- equal to which method you would start with.

00:21:33.900 -- If you want to solve the problem that you don't know

00:21:36.595 -- solution about anything about.

00:21:39.660 -- Probably oil it while it's easy to implement, its lately least

00:21:43.037 -- accurate, but it's easy to implement, and for example, if

00:21:46.107 -- you programmed at an, you see that it doesn't work. Maybe

00:21:49.484 -- there is no point of investing time, right? But if you know

00:21:53.168 -- that yes solution exists, an that gives you what you need,

00:21:56.545 -- you can start with all the method just to get a feeling of

00:22:00.536 -- what solution is going to do. But then if you need to have

00:22:04.527 -- more accuracy, or let's say if you have to compute for long

00:22:08.211 -- time and maybe. Many points then you probably would use on

00:22:12.492 -- GeForce order method. Matlab in fact has so called variable

00:22:15.782 -- Force 5th order method ricotta which allows us to change the

00:22:19.401 -- step size depending on the estimate of the error. So they

00:22:23.020 -- have some estimate of the error in air is small. Then

00:22:26.639 -- you can use largest largest step. If estimate becomes

00:22:29.600 -- large then you decrease the time step so it's not

00:22:32.890 -- constant, is not the same method that would be

00:22:35.851 -- considered here.

00:22:37.550 -- OK, I mean whatever Matlab built-in function solver.

00:22:42.500 -- OK, so an example and I'll have this available on the course

00:22:47.564 -- website and then I'll give you a hand out next time just to show

00:22:53.472 -- you what is going on in this ricotta method. So if we

00:22:58.958 -- want to solve this initial value problem starting from .12 and

00:23:03.600 -- finding oh at 1.4 in two steps using force ordering decoder

00:23:08.242 -- method, so two steps means that.

00:23:11.460 -- What is H we go from 1 to 1.4.

00:23:15.900 -- Each is.

00:23:19.750 -- So age is 1.4 -- 1 / / 2, so this will give us.

00:23:27.190 -- Zero Point 4 / 2 will be 0.2, right? So this is your step size

00:23:32.515 -- capital N number of steps is 2 inside of each step. How many

00:23:37.130 -- stages do you have?

00:23:39.330 -- Four stages right? So 4th function evaluations. So for

00:23:42.480 -- each stage you have to write K1K2K3K four and then the

00:23:46.330 -- weighted average to find next

00:23:48.080 -- approximation. So K1K2K64 will be different for

00:23:51.738 -- inside of each step.

00:23:55.390 -- OK, so H with no envy, no initial condition. X Zero is

00:23:59.626 -- one, XY0 is 2 OK, what is a function function is X + sqrt y.

00:24:04.921 -- This is your function F so F of XNYN is X N + sqrt y N.

00:24:12.780 -- OK, and then you carefully substitute these values, right?

00:24:16.263 -- I mean it's OK for demonstration purposes, so you probably want

00:24:20.520 -- to have this done by computer right? Unless function is simple

00:24:24.777 -- that you can, you can do it. OK, so gave one is a slope at first

00:24:30.969 -- point. In this case at X0Y0, right? You find Cato is you

00:24:35.613 -- March, you replace X with 0 + H to point in between and Y zero.

00:24:41.418 -- You follow The Cave one slope.

00:24:45.130 -- Right, so this is your X value. This is your why value once you

00:24:48.882 -- have them, you substitute them in the function, so you replace

00:24:51.830 -- X with this. Why is that?

00:24:54.140 -- Annual value it so this gives you slope K2 then use K2 here to

00:24:59.768 -- find .3 again. X is just half step away while zero plus K 2 *

00:25:05.798 -- H / 2 This is your ex. This is your Y value you put in the

00:25:12.230 -- function you evaluate. Finally K 4 you much full step.

00:25:16.890 -- Use slope case 3. This is your X value. This is your.

00:25:20.694 -- Why will you find slope K 4? You take weighted average.

00:25:24.181 -- You get next approximation.

00:25:31.380 -- OK, so now what you found you found.

00:25:37.130 -- X1 is 1.2 and Y one is 2.5201.

00:25:45.570 -- So now you use this.

00:25:47.540 -- To do another step so we have two steps here to do.

00:25:51.350 -- Right, so we have this and then again K1K2K3K four. But now

00:25:56.990 -- instead of X0Y0 you have X1Y1.

00:26:00.690 -- Just indexes shifted and so on, so I'll have this online and

00:26:05.034 -- I'll bring this on Monday.

00:26:09.020 -- OK, any are there any questions yes.

00:26:14.720 -- This is based on.

00:26:17.670 -- Next one you just. Right, you found this one from the previous

00:26:22.220 -- right step and then you just keep it the same, but you keep

00:26:26.640 -- adding. So what I do OK, I have formulas dependent on X&YN

00:26:30.720 -- right? So here I had to use.

00:26:34.040 -- My end was zero.

00:26:37.050 -- So I replace end with zero everywhere before I try to

00:26:41.285 -- compute anything. So in the next stage I have to use N equals.

00:26:46.910 -- 1.

00:26:48.560 -- OK so I replace.

00:26:51.160 -- SNV X1 Y end with Y1 and similarly everything else but

00:26:56.011 -- K1K2K3 will be different now from the previous case from the

00:27:00.862 -- previous step. So I have F of X1Y1 compared to.

00:27:06.930 -- F of X0Y0 I have for K2 I have F of X1 plus HY one plus K 1 * H

00:27:14.250 -- / 2 I have here with HO, but this key one and escape one of

00:27:19.740 -- the same. OK, so at every state at every step you

00:27:24.894 -- K1K2K3K four will be different, so he probably

00:27:28.142 -- technically we have to write down another index an, but

00:27:32.202 -- it just will increase. It will be very cumbersome. So

00:27:36.262 -- so all slopes are different. So for each step you

00:27:40.322 -- recompute your slopes.

00:27:44.870 -- OK, that's why.

00:27:47.280 -- Write this before you implement your substitute values.

00:27:52.260 -- OK, right X 0X1YY1Y2 and so on.

00:28:00.600 -- This will not be on the test.

00:28:04.540 -- OK, but it is in the homework so you have to do it.

00:28:10.850 -- OK, any other questions?

00:28:16.150 -- So more about numerical methods. So we teach a

00:28:20.236 -- course which is now taught between three department's

00:28:23.868 -- mathematics, physics, and engineering is typically

00:28:26.592 -- chemical genius teaching and then so this method are

00:28:30.678 -- studied in more details, but not only this, but also

00:28:35.218 -- root, finding methods, argon values, eigenvectors,

00:28:37.942 -- solving linear systems. So maybe I should write so.

00:28:47.210 -- More about.

00:28:58.720 -- Anne.

00:29:01.050 -- 428 and there's also so this physics for 28 and engineering.

00:29:07.850 -- So it is the same course. I mean, of course the also

00:29:12.602 -- graduate version.

00:29:15.760 -- 529 I think and physics.

00:29:20.070 -- 528 So it's slightly dependants who is teaching, but we cover

00:29:24.437 -- the same material, so professors from different departments POV

00:29:28.010 -- alternate, but we have the same syllabus to follow.

00:29:36.920 -- No, normally you choose whatever flavor you want on

00:29:40.664 -- your transcript, but that's the only difference.

00:29:46.810 -- OK questions.

00:29:51.800 -- So.

00:29:53.810 -- I'll start Chapter 3, which is linear equations of

00:29:57.689 -- higher order.

00:30:16.410 -- So far we've dealt only with first order linear equations,

00:30:20.940 -- but we will look at their methods that will allow us to

00:30:26.376 -- solve equations of high order and linear equations do not

00:30:30.906 -- require. Coefficients to be constantly constant, but we will

00:30:35.444 -- for simplicity we will start with questions of miss

00:30:39.656 -- constantly efficients. OK, so let's just recall the definition

00:30:43.868 -- of the linear equation of ends order so linear.

00:30:51.720 -- And order.

00:30:54.120 -- Differential equation. Has function derivative, second

00:30:58.768 -- derivative, and so on up the derivative order NPL linearly in

00:31:04.675 -- the equation so?

00:31:07.440 -- Hey Ann.

00:31:13.720 -- Plus a N -- 1.

00:31:21.670 -- Loss etc plus a 2X.

00:31:25.550 -- D2Y T X ^2.

00:31:29.040 -- Plus a one of X.

00:31:34.190 -- Plus a 0 times function Y

00:31:37.382 -- equals. Some function that does not depend on why.

00:31:44.150 -- So remember.

00:31:46.460 -- How, how, how we define linear function we defined in a

00:31:50.673 -- function is a X + B right? So your independent variable AP is

00:31:55.652 -- linearly means raised to the power one. So now in the linear

00:32:00.248 -- differential equation you have the same but for the function

00:32:04.078 -- derivative, second derivative and up to the ends of the

00:32:07.908 -- derivative. These are the functions of X only.

00:32:11.540 -- Right then they don't involve why dependence are

00:32:14.716 -- of X is right inside.

00:32:18.020 -- Can be 00 but linearity means that you don't have y ^2.

00:32:22.604 -- Don't have y * Y prime and so on so they appear linearly

00:32:27.952 -- same way as X appears in the linear function.

00:32:32.770 -- In this case, we multiply by constant in the equation. In

00:32:36.268 -- the case of, the equation, coefficients can be functions

00:32:39.130 -- of X at most.

00:32:41.960 -- OK. So if.

00:32:46.070 -- Oldest coefficients.

00:32:51.390 -- Constants.

00:32:57.190 -- Then we have equations with constant coefficients.

00:33:00.960 -- Then differential equation is.

00:33:05.460 -- A linear.

00:33:08.260 -- Differential equation with.

00:33:13.190 -- Constant.

00:33:18.560 -- Coefficients. And these are, these equations are

00:33:23.016 -- typically easier to solve. Otherwise equation has

00:33:26.103 -- variable coefficients.

00:33:36.630 -- This differential equation is.

00:33:41.680 -- Linear, viz.

00:33:48.200 -- Variable coefficients.

00:33:53.330 -- OK.

00:33:55.070 -- If you have a linear equation an if right hand

00:33:59.190 -- side is identically zero, then we have linear

00:34:02.486 -- homogeneous equation and in fact homogeneous equation

00:34:05.370 -- only can be introduced for linear equations. I mean

00:34:09.078 -- sometimes can be introduced for nonlinear, but typical

00:34:12.374 -- is for linear equations.

00:34:15.830 -- Then

00:34:19.440 -- linear differential equation.

00:34:22.060 -- Is homogeneous.

00:34:29.190 -- Otherwise.

00:34:35.270 -- Linear differential equation is.

00:34:42.490 -- Nonhomogeneous

00:34:47.710 -- let's look at some examples that we've just trying to classify

00:34:51.593 -- and then to analyze the order if it is linear. If it is

00:34:56.182 -- homogeneous or non homogeneous.

00:35:01.010 -- So Y double prime plus X y = 0. So what is the

00:35:05.716 -- order of this equation?

00:35:09.740 -- 2nd order.

00:35:12.000 -- Is it linear or nonlinear?

00:35:16.800 -- Linear right? Because XY is multiplied by a function of

00:35:21.040 -- XY, double prime is multiplied by one. So linear is a

00:35:25.704 -- sensitive linear. Is it homogeneous or non

00:35:28.672 -- homogeneous?

00:35:32.050 -- Homogeneous because there is no function that only depends on X

00:35:36.428 -- rated 0 so homogeneous.

00:35:42.100 -- Coefficients are constant or variable.

00:35:46.930 -- Variable because we have X right? So this.

00:35:55.180 -- Variable coefficients. OK.

00:35:59.730 -- What about this equation?

00:36:03.530 -- X ^2 y double prime minus two XY prime plus Y to the XY equals

00:36:10.640 -- two X -- 1.

00:36:13.550 -- Order

00:36:15.760 -- 2nd. Is it linear or nonlinear?

00:36:25.740 -- OK, so we have Y times each of the XY prime times minus two XY

00:36:30.630 -- double prime times X squared. We have termed it depend on why

00:36:34.868 -- is it in this form?

00:36:38.870 -- That you have derivatives multiplied by at most

00:36:41.350 -- functions of X.

00:36:43.760 -- Yes, so it is linear, right?

00:36:46.730 -- Is it homogeneous since it is linear or not homogeneous?

00:36:51.860 -- None, because we have to explain this one.

00:36:56.590 -- So, nonhomogeneous? And coefficients are variable

00:37:00.892 -- variable right? Because we have functions so this.

00:37:07.660 -- Variable coefficients.

00:37:12.280 -- OK, next example.

00:37:15.440 -- Is 2 Y triple prime minus three Y prime plus seven Y equals

00:37:22.499 -- luxury four X ^2 -- 1?

00:37:26.490 -- OK, the order of the equation is 3 third order.

00:37:35.940 -- Is it linear or nonlinear?

00:37:42.440 -- Huh?

00:37:43.970 -- Linear or nonlinear?

00:37:47.530 -- Why is it nonlinear?

00:37:52.610 -- We have function multiplied by 7 derivative multiplied by

00:37:57.002 -- negative three, so the order to multiply by two.

00:38:03.220 -- Linear.

00:38:07.420 -- What is in here an is 3.

00:38:10.950 -- Look for linear equation. You have function multiplied by at

00:38:14.430 -- most, so this may be

00:38:16.518 -- constant. Or maybe some function of X. This functional effects

00:38:20.159 -- may be nonlinear, but we look at the look at the YY prime Y

00:38:24.373 -- double prime up to the highest order derivative, not in terms

00:38:27.684 -- of X in terms of Y.

00:38:30.880 -- OK. So equation is.

00:38:34.260 -- Linear.

00:38:36.670 -- Since it is linear, is it homogeneous or homogeneous?

00:38:41.910 -- Non, because of the logarithm of X ^2. So nonhomogeneous

00:38:46.250 -- and coefficients are.

00:38:48.580 -- Constant rate with constant coefficients.

00:38:54.990 -- OK and last example.

00:38:59.640 -- White triple prime my plus 2Y double prime

00:39:04.216 -- minus y * Y prime +7.

00:39:08.920 -- The order is.

00:39:11.750 -- So the order. 3rd order.

00:39:16.710 -- Linnaean olenia.

00:39:21.560 -- Nonlinear because we have y * y prime right nonlinear.

00:39:28.920 -- We cannot say if it is ominous nonhomogeneous because we don't

00:39:33.980 -- have linearity to say this.

00:39:38.200 -- OK.

00:39:40.660 -- So big chunk of this course will be devoted on the 2nd order well

00:39:45.924 -- probably not sister going to order, so essentially it's

00:39:49.308 -- easier probably to solve 2nd order equations, especially when

00:39:52.692 -- you consider with variable coefficients. But the method

00:39:55.700 -- that we will develop for equations with constant

00:39:58.708 -- coefficients can be easy.

00:40:00.470 -- Applied to the 2nd order first Order 5th order intense order I

00:40:06.086 -- will have 19th order example to consider. So yes.

00:40:15.070 -- It is defined only for linear for linear equations, so.

00:40:21.530 -- I've seen some definitions that say if identical is zero

00:40:24.950 -- solution satisfies equation, then you can think of this as

00:40:28.370 -- homogeneous. In this case it won't be because if you have

00:40:32.132 -- zero then this is 0. This is non 0 but typically homogeneous is

00:40:36.578 -- only for linear equations because you have some relation

00:40:39.656 -- to linear algebra. So linear systems, linear equations so

00:40:42.734 -- that's the reason. So once you may have a question on the

00:40:47.180 -- test to classify equation equations and then so similar

00:40:50.258 -- like we we've done here.

00:40:52.400 -- You look at the order if it is linear then you can think

00:40:56.716 -- it's homogeneous, nonhomogeneous, but if it's

00:40:58.708 -- not linear then you just stop.

00:41:01.720 -- OK.

00:41:06.820 -- OK, so let's start with second order linear

00:41:10.396 -- homogeneous equations so.

00:41:15.190 -- So we consider 2nd.

00:41:19.420 -- Modern.

00:41:24.600 -- Linear homogeneous differential equations.

00:41:30.280 -- We will first address the problem when we have none of

00:41:34.031 -- them, we have homogeneous equation. Once we know how

00:41:37.100 -- to solve homogeneous then we will study how to solve

00:41:40.510 -- nonhomogeneous equations because there are different

00:41:42.556 -- methods how to address this problem. OK, so in general,

00:41:45.966 -- if you have second order linear equation then you can

00:41:49.376 -- write it in just using some coefficients which are

00:41:52.445 -- functions of X.

00:41:55.200 -- A1 of X.

00:41:57.440 -- Divide the X + A zero XY homogeneous. This means very

00:42:03.776 -- inside is 0.

00:42:15.190 -- And so let's look at example and then we will try to establish

00:42:20.546 -- some properties of solutions to the homogeneous equations.

00:42:25.750 -- So example is.

00:42:38.500 -- Let's let's let's do 2 examples, so example a.

00:42:43.530 -- X ^2 D two YG X ^2 -- 2 X divided X.

00:42:51.680 -- Plus plus two y = 0.

00:42:54.760 -- So you can see it is second order, right?

00:42:58.620 -- It is linear 'cause you have y * 2 divided you exams minus 2X and

00:43:03.480 -- this is also linear term and it is not just homogeneous because

00:43:07.368 -- there is no function that only depends on X and not multiplied

00:43:11.256 -- by wire derivative and.

00:43:13.310 -- My first statement is that the X ^2.

00:43:17.680 -- Is a solution of this equation.

00:43:21.770 -- How do we? How do we verify that this function is a solution?

00:43:27.300 -- We have the substitute and check if you get identity right. OK,

00:43:30.840 -- So what do we have? If X squared is a solution, what is the

00:43:34.970 -- derivative of this solution?

00:43:37.500 -- 2X and 2nd derivative will be 2, so we have X ^2 * 2 -- 2

00:43:44.572 -- X times. Two X + 2 times function. So do we have 0?

00:43:51.430 -- We have two X ^2 -- 4 X squared plus two X squared

00:43:55.863 -- right, so cancels so 0 = 0. So this means that X squared

00:44:00.296 -- is a solution of the equation. What happens if we?

00:44:05.020 -- Multiply this function by constant.

00:44:09.370 -- By some arbitrary constant.

00:44:13.130 -- The claim is that this is also a solution.

00:44:18.740 -- So C One is an arbitrary constant.

00:44:25.870 -- Indeed.

00:44:27.990 -- 1st Order derivative will be 2 C 1X and 2nd order derivative will

00:44:32.839 -- be 2C1, right?

00:44:35.290 -- So we have X ^2 * 2 C 1 -- 2 X times 2C. One X

00:44:43.162 -- + 2 * y C One X ^2.

00:44:48.260 -- C1 is present in all the terms, right and otherwise

00:44:51.760 -- you have two X ^2 -- 4 X squared X squared, so this

00:44:56.660 -- is also zero. So again, if you take a solution of a

00:45:00.860 -- linear homogeneous equation multiplied by arbitrary

00:45:02.960 -- constant, you still get the solution, so this will be

00:45:06.460 -- still a solution.

00:45:08.850 -- So similarly.

00:45:11.370 -- And you can verify that X is a solution.

00:45:18.240 -- The first derivative is.

00:45:20.930 -- One second derivative is 0, right? So we have X ^2 * 0

00:45:26.819 -- plus. I'm sorry minus.

00:45:32.040 -- Minus two X * 1 + 2 times function you can see that

00:45:37.318 -- this is 0.

00:45:40.060 -- And if I multiply this solution by an arbitrary constant, I also

00:45:44.452 -- get a solution.

00:45:49.450 -- Let's say C 2 * X is a solution.

00:45:55.110 -- And we can verify this by substitute and so again, second

00:45:58.883 -- derivative will be 0, so we have X, y ^2 * 0 -- 2 X times C 2

00:46:05.057 -- + 2 * C Two X.

00:46:07.860 -- Zero and finally, if you consider linear combination of

00:46:12.225 -- these two functions.

00:46:14.860 -- In linear combination is you multiply function by constant by

00:46:19.410 -- different constant and you add

00:46:21.685 -- so C1. X ^2 + C two X is.

00:46:27.720 -- Also a solution.

00:46:33.150 -- OK, let's let's verify, because probably those cases are easy to

00:46:36.560 -- see. This one is a little bit tricky. OK, so we have X squared

00:46:40.900 -- times second derivative. What is the 2nd derivative here?

00:46:45.460 -- To see one right plus zero.

00:46:48.830 -- Minus two X times first order

00:46:51.908 -- derivative 2C1X. Plus C2.

00:46:56.020 -- And plus two times functions, so C One X ^2.

00:46:59.830 -- Plus it 2X.

00:47:02.870 -- Do we have here?

00:47:06.240 -- So if I look at terms with C1.

00:47:10.270 -- I have two X ^2 -- 4 X squared, two X squared, they cancel.

00:47:18.040 -- In terms with C2.

00:47:21.470 -- Minus two XY2 plus to exit to

00:47:24.837 -- also cancel. Right, and this is here.

00:47:29.070 -- So 0 = 0.

00:47:36.510 -- So what we showed here is that if you have linear homogeneous

00:47:41.178 -- equation an if you have solutions, you form linear

00:47:44.679 -- combination. So you multiply by constants and you add and you

00:47:48.958 -- have you keep them arbitrary. Then result is also a solution

00:47:53.237 -- to this equation.

00:48:01.930 -- So maybe just another example be.

00:48:05.740 -- G2Y G X ^2 +

00:48:09.486 -- 3. Divide X + 2 * y = 0 again. This is second

00:48:17.322 -- order equation. Linear homogeneous coefficients are.

00:48:22.880 -- Constant variable so 2nd order.

00:48:29.390 -- Linear homogeneous.

00:48:34.050 -- With constant coefficients.

00:48:39.860 -- And the claims here are that E to the minus X is a solution. So

00:48:44.615 -- at this point I'm not saying how we find them, we will. We will

00:48:49.053 -- know this soon, but let's just check. So if you have it to

00:48:53.491 -- the minus X derivative will be minus E to the minus X second

00:48:57.612 -- derivative will be with the plus sign, right? So you have either

00:49:01.416 -- the minus X + 3 * E to the minus X minus sign plus two times

00:49:06.488 -- function E to the minus X.

00:49:10.010 -- You get 0 right, and similarly if you multiply by constant.

00:49:16.900 -- Is a solution.

00:49:20.190 -- That I will not verify, but you can see that this is also

00:49:23.284 -- straightforward to do.

00:49:27.080 -- And then another solution here available is E to the minus, 2X

00:49:32.864 -- is a solution.

00:49:38.360 -- And if we multiply by constant, it is a -- 2. X is a solution.

00:49:45.230 -- And finally, linear combination is.

00:49:50.340 -- Also a solution.

00:49:53.260 -- OK.

00:49:56.540 -- So the result is how much time do I have left?

00:50:05.830 -- One minute. OK, so I'll I'll write the just result

00:50:09.690 -- so theorem.

00:50:12.320 -- So principle.

00:50:15.760 -- Of linear superposition?

00:50:22.460 -- It only works for linear homogeneous equations, so given.

00:50:28.790 -- 2nd order equation.

00:50:39.160 -- 2nd order.

00:50:42.280 -- Linear.

00:50:44.550 -- Homogeneous equation.

00:50:51.200 -- If. Why one of XY2 of X?

00:50:56.650 -- Our solutions.

00:51:01.250 -- Of this differential equation.

00:51:05.810 -- Then

00:51:08.050 -- their linear combination.

00:51:15.580 -- C1Y one of X + y two Y2FX is also solution

00:51:21.212 -- of the same equation.

00:51:37.400 -- See once you're here.

00:51:41.630 -- C1C2 are arbitrary constants.

00:51:50.360 -- And similar result holds 4th order equations, right? So this

00:51:54.430 -- doesn't change. OK, so I guess I'm out of time, any questions?

00:52:01.690 -- OK, thank you and drive safely.

Duration:"00:56:27.8400000"

00:00:21.260 -- Alright, good morning.

00:00:22.202 -- Welcome to class city.

00:00:23.460 -- You got your homework turned in?

00:00:25.340 -- How was the assignment? Do OK with it.

00:00:29.170 -- How was the airfoil problem the long?

00:00:34.050 -- But solvable.

00:00:34.688 -- Just kind of have to be patient

00:00:36.921 -- as you work your way through it.

00:00:39.240 -- OK, so since you're here right now,

00:00:40.970 -- I will tell you next time we

00:00:42.503 -- have a midterm next Thursday.

00:00:43.930 -- We have mentored.

00:00:45.007 -- There will be an airfoil problem on the exam.

00:00:48.200 -- So study that up.

00:00:49.276 -- We'll review it a little bit later on today,

00:00:51.830 -- OK? Alright I have.

00:00:54.616 -- I have colleagues here at the university

00:00:57.292 -- and some friends in the law school.

00:00:59.620 -- That's probably a bad sign,

00:01:01.420 -- right that I have lawyer friends.

00:01:05.130 -- And I always ask, you know,

00:01:07.130 -- do you do you bring up current

00:01:09.265 -- events in your classes?

00:01:10.790 -- You know they're always the Supreme

00:01:12.854 -- Court litigation stuff going on?

00:01:14.450 -- I hear no all the time.

00:01:16.450 -- It takes too much work.

00:01:18.120 -- We do not do that in gas tax.

00:01:20.780 -- OK, just to be clear,

00:01:22.450 -- so.

00:01:25.150 -- This is. The seven minutes of terror.

00:01:29.460 -- OK, the perseverance Lander

00:01:30.888 -- that landed on Mars last week.

00:01:33.030 -- So I thought that I would go through

00:01:35.854 -- step by step to let you know what's

00:01:38.812 -- going on and then watch a video of it.

00:01:41.950 -- Just 'cause I'm the teacher, right?

00:01:44.096 -- And I say we could watch videos,

00:01:46.590 -- we're going to watch videos, all right.

00:01:49.092 -- Let's check this out.

00:01:50.520 -- So 10 minutes before landing,

00:01:52.300 -- the shell comes off and this

00:01:54.346 -- is what the Lander looks like,

00:01:56.590 -- and you can see that it maneuvers itself.

00:01:59.850 -- OK, now it's 8 minutes to entry.

00:02:01.870 -- Gets balance right here.

00:02:02.958 -- What kind of body would we call that right

00:02:05.550 -- there, that kind of rounded surface?

00:02:09.750 -- That's a blunt body. OK,

00:02:12.520 -- that's how that's how any kind of spaceship.

00:02:18.460 -- Enters the atmosphere with a blunt body and

00:02:21.100 -- we'll see that once it starts to enter.

00:02:23.630 -- Look at this, although they

00:02:25.170 -- don't draw it as a shock.

00:02:27.180 -- We know because we are experienced

00:02:29.352 -- gas dynamicist that is a

00:02:31.554 -- shockwave that occurs right there

00:02:33.274 -- and notice that's a peak heating.

00:02:35.290 -- OK, let's look at some of these numbers.

00:02:39.140 -- Here it is 78 miles in altitude

00:02:42.262 -- above the Martian surface.

00:02:44.430 -- Look at this velocity 13,000.

00:02:47.670 -- Mph. Let's move in.

00:02:52.573 -- That is about 5900 meters per second, OK?

00:03:00.730 -- And let's see here.

00:03:02.322 -- So I did a little bit of calculation here.

00:03:06.760 -- The Martian atmosphere temperature.

00:03:08.512 -- Obviously it ranges like

00:03:10.264 -- every other atmosphere,

00:03:11.870 -- but kind of a standard temperature

00:03:15.272 -- is 60 degrees Centigrade below 0.

00:03:18.410 -- Minus 60 pretty cold, so that is 213

00:03:22.210 -- Kelvin enters at 5900 meters per second.

00:03:25.730 -- It's Martian atmosphere is primarily

00:03:28.170 -- carbon dioxide is about 95% carbon dioxide,

00:03:31.548 -- and so it's gamma is about 1.3

00:03:34.831 -- and its molecular weight is 44.

00:03:37.930 -- So we can calculate RA 314 divided

00:03:41.038 -- by 44 and it works out 'cause you

00:03:44.993 -- know how to calculate Mach numbers.

00:03:48.430 -- This works out to be a Mach number of 25.6.

00:03:53.910 -- Entering the atmosphere.

00:03:58.570 -- That's moving. That's moving OK,

00:04:00.990 -- so one of the one of the one of the

00:04:03.975 -- primary designs for this blunt body is that

00:04:07.951 -- you're traveling on Mach number of 25.6.

00:04:10.940 -- Tell me do parachutes work

00:04:12.895 -- very well in a lot #25.6 no.

00:04:15.720 -- OK, we will talk actually a little

00:04:18.303 -- bit later on in the semester that

00:04:21.516 -- there are some supersonic parachutes.

00:04:24.050 -- OK, there are some,

00:04:25.506 -- but this is not designed to do that.

00:04:28.390 -- So what the blunt body does is

00:04:30.721 -- that it absorbs that kinetic

00:04:32.582 -- energy and slows it down.

00:04:34.550 -- OK, so it's moving that amount #25.

00:04:37.080 -- Here's where it heats up that peak

00:04:39.460 -- heating and then it starts to decelerate,

00:04:42.150 -- and then finally finally it starts

00:04:44.244 -- to slow down and what it can do

00:04:47.223 -- this hypersonic aero maneuvering

00:04:48.731 -- what it does is kind of vibrate.

00:04:51.200 -- So it goes like this and it

00:04:53.657 -- converts that kinetic energy into.

00:04:55.650 -- Thermal energy and slows down.

00:04:57.510 -- That's what this maneuvering is.

00:04:59.360 -- And then finally.

00:05:00.803 -- Begins to deploy a parachute.

00:05:03.210 -- Does that at 400 meters per second.

00:05:05.980 -- Now, if you think in air that's

00:05:08.696 -- a little over Mach one,

00:05:10.730 -- it turns out for the Martian atmosphere and

00:05:14.098 -- these times it's a little bit over Mach one,

00:05:17.470 -- so it turns out that this supersonic,

00:05:20.240 -- but not real high,

00:05:21.820 -- not like hypersonic.

00:05:23.010 -- A parachute begins to deploy,

00:05:24.990 -- then the heat shield that

00:05:26.970 -- protected the spacecraft.

00:05:28.160 -- Right here, drops off.

00:05:29.988 -- OK falls off and then over here.

00:05:33.290 -- This is kind of neat.

00:05:35.210 -- The shell drops off and then there's

00:05:37.975 -- a little radar system right there

00:05:40.454 -- in that beams down to the surface to

00:05:43.756 -- determine the best place to land.

00:05:46.150 -- So it's like an autopilot.

00:05:48.140 -- It's like it's like an auto,

00:05:50.530 -- an artificial intelligence system.

00:05:52.602 -- That search arounds figures

00:05:54.674 -- out the best place to land OK?

00:05:57.420 -- Starts to collect that data and

00:05:59.706 -- then this shell drops off right

00:06:02.030 -- here and the Lander starts to

00:06:04.190 -- descend to starts to descend.

00:06:06.280 -- Excuse me, comes down,

00:06:07.776 -- it's got 4 little rockets right here

00:06:10.480 -- and these are called cold gas thrusters.

00:06:13.210 -- That's pressurized gas there,

00:06:14.902 -- so it's not necessarily a combustion

00:06:17.507 -- process starts to descend right here and

00:06:20.020 -- then it deploys what's called a sky crane,

00:06:22.830 -- so this hovers right here with those rockets,

00:06:25.910 -- and what the sky crane does.

00:06:28.380 -- Is that it lowers the Lander CABI

00:06:30.606 -- wires so that the Lander lands,

00:06:32.790 -- and then it cuts those wires in the sky,

00:06:35.840 -- crane falls away and you have

00:06:37.730 -- soft landing on the surface?

00:06:41.890 -- First off, if you're an engineer,

00:06:43.400 -- you can't look at that and say

00:06:45.213 -- that is not absolutely awesome.

00:06:47.230 -- To see that in action OK Now is actually.

00:06:49.980 -- That's kind of a complicated process if you

00:06:52.188 -- think of everything that's going on there,

00:06:54.550 -- you know one of the few seniors here that

00:06:57.016 -- learn about the Kiss principle, right?

00:06:59.125 -- Keep your design simple so that they work.

00:07:01.570 -- This is actually a pretty

00:07:03.090 -- complicated process.

00:07:03.700 -- Past Mars Landers.

00:07:06.030 -- Past Mars Landers actually have a

00:07:08.022 -- kind of a balloon on the outside

00:07:10.452 -- number of balloons that cover this in,

00:07:12.870 -- so that when it lands,

00:07:14.580 -- it actually bounces on the

00:07:16.230 -- surface of Mars until it stops.

00:07:18.340 -- Then the balloons deflated, opens up.

00:07:20.390 -- Then you have a Lander,

00:07:22.100 -- so spirit and opportunity landed like that.

00:07:24.500 -- OK, so.

00:07:26.460 -- That the pretty complicated

00:07:27.820 -- sort of landing process.

00:07:29.180 -- Now let's watch the video.

00:07:30.880 -- They actually in fact I heard

00:07:32.896 -- this on the news is that is that

00:07:35.661 -- in part of the design process,

00:07:37.680 -- they told a couple of engineers say hey,

00:07:40.400 -- listen,

00:07:40.804 -- would it be kind of cool to take high def?

00:07:44.890 -- Pictures,

00:07:45.179 -- High resolution pictures of the

00:07:46.624 -- landing process so they actually went.

00:07:48.450 -- I want to see RadioShack that kind

00:07:50.585 -- of dates me so they went to off

00:07:53.100 -- the shelf cameras and were able to

00:07:55.232 -- put it on the Lander so that you

00:07:57.360 -- can see the landing in process.

00:08:00.400 -- Dumb question, would you like to see it?

00:08:03.570 -- I thought so, so here we go.

00:08:05.630 -- So now that you know.

00:08:07.100 -- So now that you know the landing process,

00:08:09.450 -- let's see what we got here and fly

00:08:11.482 -- right maneuver where the spacecraft

00:08:13.007 -- will jettison the entry balance

00:08:14.612 -- masses in preparation for parachute

00:08:16.198 -- deploy and to roll over to give the

00:08:18.563 -- radar a better look at the ground.

00:08:23.330 -- Public it indicates she's deployed.

00:08:26.610 -- The navigation has confirmed that

00:08:28.390 -- the parachute has deployed an.

00:08:30.170 -- We're seeing significant

00:08:31.235 -- deceleration in the velocity.

00:08:32.660 -- Our current velocity is 450 meters

00:08:34.658 -- per second at an altitude of about 12

00:08:37.545 -- kilometres from the surface of Mars.

00:08:42.310 -- He tilts up. Perseverance is now,

00:08:45.000 -- so she was dropping off these

00:08:47.010 -- and then literally on Mars.

00:08:48.750 -- Just allow both the radar and the

00:08:50.920 -- cameras to get their first look

00:08:52.991 -- at the surface current velocity

00:08:54.771 -- it 145 meters per second at an

00:08:57.263 -- altitude of about 10 Columbia 9

00:08:59.321 -- 1/2 kilometers above the surface.

00:09:03.160 -- Now that's pretty cool to see that close.

00:09:06.420 -- Picture in that sort of definition of

00:09:08.667 -- the Martian surface. That's pretty neat.

00:09:14.260 -- And if you look on the bottom,

00:09:16.470 -- you can kind of see it's.

00:09:18.370 -- It gives the step filter contract about

00:09:20.540 -- players entry .3 meters that there

00:09:22.546 -- should come out altitude 7.4 kilometers

00:09:24.574 -- now has radar lock on the ground.

00:09:26.580 -- Current city is about 100 meters per second,

00:09:29.110 -- 6.6 kilometres of the surface.

00:09:35.220 -- President is continuing to

00:09:36.888 -- descend on the parachute.

00:09:38.560 -- We're coming up on the initialization of

00:09:41.507 -- terrain relative navigation and subsequently

00:09:43.641 -- the priming of the landing engines.

00:09:46.060 -- Our current velocity is about 90 meters per

00:09:49.724 -- second at an altitude of 4.2 kilometers.

00:09:53.320 -- Now, whether or not your geologist,

00:09:55.090 -- you could look at that surface

00:09:56.692 -- and say that there's windblown.

00:09:58.340 -- You can see the guys mentioned

00:10:00.086 -- that the reservation system has

00:10:01.595 -- produced a valid solution here,

00:10:03.060 -- and part of strain out the navigation water

00:10:05.500 -- or liquid that was there be a nominal.

00:10:07.780 -- We have timing of the landing engine's.

00:10:15.160 -- Back Shell survival at sea is 83

00:10:17.589 -- meters per second at about 2.6

00:10:19.838 -- kilometers from the surface of Mars,

00:10:22.110 -- we have confirmation that the

00:10:23.940 -- back shell has separated.

00:10:25.410 -- We are currently performing

00:10:26.870 -- the divert maneuver.

00:10:27.970 -- Travelocity is about 75 meters per

00:10:30.256 -- second at an altitude of about a

00:10:32.874 -- kilometre off the surface of Mars.

00:10:34.920 -- Here, in safety Bravo.

00:10:38.090 -- We have completed our

00:10:39.770 -- terrain relative navigation.

00:10:41.030 -- Current speed is about 30

00:10:43.130 -- meters per second altitude,

00:10:44.810 -- about 300 meters off the surface of Mars.

00:10:50.680 -- We have started our constant

00:10:52.570 -- velocity accordion, which means

00:10:54.084 -- we are conducting the skycrane,

00:10:55.970 -- so this is where the sky Crane

00:10:58.616 -- maneuver deploys at it drops,

00:11:00.510 -- there's the Lander right there?

00:11:02.400 -- I mean, that's that is totally cool,

00:11:05.040 -- 20 meters off the surface.

00:11:13.220 -- And you see that in the top view,

00:11:15.220 -- that's the that's the sky cream.

00:11:16.720 -- Rocky go delta.

00:11:18.148 -- Captain confirmed persevered

00:11:19.576 -- safely on the surface of Mars,

00:11:22.130 -- ready to begin seeking

00:11:23.930 -- the sands of past life.

00:11:29.220 -- There you go.

00:11:32.380 -- How great is that?

00:11:33.828 -- And that is pretty cool.

00:11:35.640 -- And since you are,

00:11:37.088 -- since you were all gas dynamicist's,

00:11:39.260 -- you know you could figure out the

00:11:42.186 -- aerodynamics of a good portion

00:11:44.444 -- of everything that we saw there.

00:11:47.210 -- In a couple of weeks actually,

00:11:49.100 -- starting next week,

00:11:50.528 -- we'll learn about rocket nozzles.

00:11:52.910 -- OK, and how they work good.

00:11:56.780 -- Any questions at all for get going?

00:11:58.950 -- Is it not a beautiful day

00:12:00.600 -- for gas at the office today?

00:12:02.670 -- It is awesome, OK?

00:12:05.420 -- Thursday is an exam at the end of class.

00:12:08.480 -- Today we will have a little review

00:12:11.360 -- an what's going to be on the exam,

00:12:13.920 -- but I want to just rehash

00:12:15.780 -- expansion waves to make sure

00:12:17.462 -- you've got that down expansion,

00:12:19.360 -- which are very important concept.

00:12:21.400 -- OK so.

00:12:23.500 -- Recall that an expansion wave

00:12:25.810 -- occurs when you have a supersonic

00:12:28.231 -- flow that turns away from itself,

00:12:30.650 -- so it's coming down this way

00:12:32.828 -- known as the mod numbers.

00:12:35.010 -- One goes through goes through

00:12:36.995 -- a turn away from itself.

00:12:38.980 -- There are some questions

00:12:40.568 -- last time after class,

00:12:42.160 -- so I just want to clarify

00:12:44.542 -- this that this line,

00:12:46.130 -- this leading Mauchline in this

00:12:48.205 -- trailing Mauchline constitute

00:12:49.450 -- what's called an expansion fan.

00:12:51.290 -- OK, those are expansion waves that.

00:12:53.810 -- That occur when a supersonic

00:12:56.005 -- flow turns away from itself.

00:12:58.200 -- OK, this leading Mauchline,

00:12:59.992 -- then that's the front edge of that fan,

00:13:03.470 -- and it occurs at this Mach angle.

00:13:06.540 -- Musa one OK?

00:13:09.470 -- It's it's the air turns through this

00:13:11.990 -- fan once it leaves the fan then it

00:13:14.769 -- falls the wall as it comes down here OK.

00:13:17.890 -- In the turning angle Delta OK,

00:13:20.590 -- recall a number of things.

00:13:22.840 -- The flow is isentropic.

00:13:24.640 -- No losses.

00:13:25.540 -- That means the stagnation pressure

00:13:27.790 -- remains constant across that turn,

00:13:30.040 -- their model number goes up.

00:13:33.190 -- It accelerates,

00:13:33.940 -- in fact,

00:13:34.690 -- I mentioned nozzles rocket nozzles will learn

00:13:37.357 -- how expansion waves work in rocket nozzles,

00:13:39.870 -- and that's how they that's how a

00:13:42.201 -- nozzle can produce supersonic flow.

00:13:44.320 -- Since the model number goes up,

00:13:46.550 -- the pressure goes down,

00:13:48.030 -- static pressure goes down,

00:13:49.510 -- static temperature goes down,

00:13:51.350 -- so these temperature and pressure ratios

00:13:54.182 -- right here are both less than one.

00:13:56.420 -- OK, again,

00:13:57.056 -- as you're going through problems,

00:13:58.650 -- make sure that you've got that.

00:14:01.850 -- That when you calculate those numbers

00:14:03.602 -- that your pressure and temperatures are

00:14:05.476 -- all trending in the right directions.

00:14:07.450 -- OK, if you haven't seen it already

00:14:09.445 -- in your homework problems,

00:14:10.870 -- but I've seen it all the time in exams.

00:14:13.670 -- You're under the pressure.

00:14:14.878 -- You gotta get the problem solved and

00:14:17.060 -- you accidentally invert a number.

00:14:18.640 -- Or you read it wrong.

00:14:20.200 -- OK,

00:14:20.540 -- check and make sure that all those

00:14:22.920 -- pressure temperature Mach number

00:14:24.332 -- values are going in the right direction.

00:14:26.670 -- OK.

00:14:28.710 -- Good we showed then what the

00:14:31.032 -- Prandtl Meyer angle was and it is.

00:14:33.400 -- It's a fictitious angle.

00:14:34.828 -- OK so it's not like as an angle.

00:14:37.740 -- You can take your protractor

00:14:39.540 -- out and measure it in the flow.

00:14:42.070 -- It's a mathematical construct that occurs

00:14:44.590 -- for every Mach number greater than one.

00:14:47.480 -- OK,

00:14:47.824 -- and as you learn in your in your homework

00:14:51.004 -- that you could go through the book

00:14:53.573 -- or go through com prop and be able

00:14:56.444 -- to figure out what that value of

00:14:58.840 -- Theta is that parental Meyer angle.

00:15:00.970 -- And this is the big relationship right here.

00:15:04.580 -- That the turning angle.

00:15:06.292 -- Right there is related to the

00:15:08.946 -- difference in the parental Myer angles.

00:15:12.100 -- OK. Alright, and this is this is

00:15:14.564 -- the key relationship right here.

00:15:16.420 -- When you're solving expansion weight

00:15:17.885 -- problems, you know any two values.

00:15:19.650 -- If you know the downstream Mach

00:15:21.360 -- number and the turning angle,

00:15:22.870 -- you can get the upstream Mach number.

00:15:24.920 -- If you know the upstream Mach number

00:15:26.901 -- in the downstream Mach number,

00:15:28.440 -- you get the turning angle.

00:15:29.900 -- If you know the turning angle

00:15:31.562 -- in the upstream Mach number,

00:15:33.120 -- you can get announcement number.

00:15:34.590 -- OK, so you'll be able to deduce

00:15:36.522 -- two pieces of information and

00:15:38.168 -- from this relationship right here

00:15:40.043 -- you can get the third OK.

00:15:42.030 -- Straight forward there.

00:15:46.230 -- OK, shock expansion theory.

00:15:48.054 -- You use this when you solve problems, OK?

00:15:54.360 -- So again, I don't want you to use

00:15:57.120 -- the airfoil function on com prop.

00:15:59.500 -- I want you to be able to look at

00:16:01.831 -- an airfoil like this and determine

00:16:04.504 -- if you have oblique shockwaves.

00:16:06.840 -- If you have expansion ways all

00:16:09.420 -- based on the geometry of the turn

00:16:12.395 -- right here and the angle of attack.

00:16:15.120 -- OK, let's let's, let's talk about.

00:16:17.160 -- Let's talk about a problem here real

00:16:19.505 -- quickly to make sure you've got this down.

00:16:22.260 -- 'cause this is, this is an issue

00:16:24.514 -- that comes up many times here.

00:16:26.680 -- Let's say you have a triangular

00:16:28.720 -- shaped airfoil.

00:16:29.400 -- OK, and then we'll just put

00:16:31.494 -- this at an angle of attack of 0.

00:16:34.160 -- So looks like this.

00:16:35.520 -- It's a symmetric airfoil.

00:16:39.930 -- Looks like this here and you have some

00:16:42.314 -- Mach number that's greater than one.

00:16:44.400 -- And let's say that we have a

00:16:46.647 -- turning angle here of 10 degrees

00:16:48.601 -- just to pick a number, OK.

00:16:50.590 -- K alpha is equal to 0,

00:16:53.290 -- so zero angle of attack there.

00:16:55.290 -- What kind of waveform do

00:16:56.955 -- you see at the bottom?

00:17:01.400 -- Nothing. What's the pressure that

00:17:03.725 -- acts on the bottom right there?

00:17:06.620 -- It's whatever your that's greater than one.

00:17:08.730 -- It's whatever your piece

00:17:09.934 -- of one is right there.

00:17:11.440 -- Whatever the atmospheric pressure is.

00:17:13.770 -- OK. Right, what do you see on

00:17:17.368 -- this top surface right up here?

00:17:21.580 -- Well, big shockwave.

00:17:22.555 -- The flow is turning into itself.

00:17:24.510 -- Changes 10 degrees,

00:17:25.740 -- so in oblique shock.

00:17:27.380 -- Forms right there and what's

00:17:29.240 -- the direction of the flow?

00:17:31.100 -- Flow follows the wall.

00:17:34.730 -- So it goes up this way.

00:17:36.790 -- OK, so here's where the here's

00:17:38.680 -- where the messed up part happens.

00:17:40.830 -- What occurs around the turn here?

00:17:44.510 -- Expansion wave.

00:17:46.340 -- OK, here's the question.

00:17:47.572 -- What is the turning angle

00:17:49.112 -- across the top there?

00:17:53.090 -- For me. 20 degrees.

00:17:59.000 -- Everybody see why that is.

00:18:01.140 -- If not, let's talk about it.

00:18:03.400 -- OK, if this is where the mistake comes in.

00:18:06.970 -- If it were 10 degrees.

00:18:09.810 -- Then what would be the direction

00:18:11.904 -- of the flow coming out here?

00:18:13.990 -- It would be horizontal.

00:18:16.970 -- If that turning angle 10 degrees

00:18:18.938 -- 'cause it goes down this way goes up

00:18:21.458 -- 10 degrees and so it's going to turn

00:18:23.919 -- another 10 degrees just to go horizontal.

00:18:26.330 -- OK, however the flow is not

00:18:28.772 -- horizontal on that side.

00:18:30.400 -- It goes down this way another 10 degrees.

00:18:34.230 -- OK, so this turning angle.

00:18:37.910 -- Right there. 20 degrees.

00:18:42.960 -- OK. Alright.

00:18:46.690 -- OK, you can calculate the

00:18:48.475 -- pressure in this region.

00:18:49.910 -- You can calculate the pressure

00:18:51.700 -- on this plate right here?

00:18:53.490 -- Can you calculate the pressure

00:18:55.280 -- in this region right here?

00:18:57.070 -- Absolutely OK with those three pressures.

00:18:59.910 -- Determine what the force is.

00:19:02.860 -- And then some of the forces.

00:19:05.930 -- In the vertical.

00:19:08.090 -- And the horizontal directions

00:19:09.534 -- get the lift in the draft.

00:19:11.700 -- OK.

00:19:12.160 -- Good.

00:19:12.620 -- OK, this is not so critical in

00:19:15.840 -- your calculations of lift and drag,

00:19:19.470 -- but an actually an oblique

00:19:22.250 -- shockwave forms here.

00:19:23.920 -- And the reason that is is because

00:19:26.188 -- the flow now comes down this way

00:19:28.809 -- and it's going to turn horizontally.

00:19:31.180 -- Across here.

00:19:31.922 -- So the float turns into itself,

00:19:34.150 -- and so there's going to be

00:19:35.890 -- an oblique shockwave,

00:19:36.760 -- but you don't have to worry about that

00:19:38.864 -- for to calculate the lift and drag.

00:19:41.090 -- OK, so it's this turning angle right there.

00:19:44.010 -- That causes problems.

00:19:45.102 -- I wanna make sure you got that down OK?

00:19:48.280 -- Excellent.

00:19:50.460 -- OK.

00:19:52.370 -- Overhead,

00:19:52.728 -- alright,

00:19:53.086 -- so actually all this is just a

00:19:55.592 -- review of what we talked about here

00:19:57.736 -- that the lift is equal to the sum

00:20:00.230 -- of the vertical components of all

00:20:02.150 -- the forces that act on the plate.

00:20:06.740 -- Very good. The lift coefficient

00:20:08.890 -- often used in the in determining

00:20:11.560 -- what kind of airfoil you want to

00:20:14.570 -- use and over what flight regimes.

00:20:17.590 -- OK is just the lift lift force.

00:20:21.340 -- Divided by 1/2 rho V squared,

00:20:23.660 -- this is the dynamic pressure.

00:20:26.230 -- Multiplied by the.

00:20:28.420 -- Multiplied by S here,

00:20:31.340 -- that is the area of the wing.

00:20:35.170 -- That's what S is there.

00:20:36.980 -- OK, and we showed in class in

00:20:39.101 -- the past that 1/2 rho V squared

00:20:41.245 -- is the same as one half P Mach

00:20:43.857 -- number squared right there.

00:20:47.600 -- OK.

00:20:50.040 -- So let's look at some.

00:20:51.890 -- Let's look at some things.

00:20:53.730 -- Let's let's just solve for L for

00:20:56.999 -- this relationship right here.

00:20:58.910 -- L and let's see what we can

00:21:01.101 -- do to generate lift here. OK.

00:21:03.381 -- We could have a better lift coefficient.

00:21:06.960 -- OK, so that's going to depend.

00:21:08.720 -- That's going to depend on

00:21:10.190 -- the shape of the airfoil.

00:21:11.660 -- OK, we could get better lift if you

00:21:14.212 -- have a higher freestream pressure.

00:21:16.990 -- OK, are you higher in the atmosphere,

00:21:19.430 -- lower in atmosphere that governs that?

00:21:22.880 -- But look at this right here. The lift.

00:21:26.392 -- Goes like the Mach number squared.

00:21:30.870 -- So what does that say about how fast you fly?

00:21:33.580 -- What is that going to generate?

00:21:36.120 -- More. Lift.

00:21:39.910 -- OK, so that means you can have

00:21:42.570 -- some very poorly shaped airfoils.

00:21:45.090 -- OK, so this Isabelle.

00:21:46.890 -- This lift coefficient might be bad.

00:21:49.590 -- But if your Mach number is high enough.

00:21:52.050 -- You can generate a lot of lift.

00:21:54.650 -- My number squared.

00:21:56.045 -- Also says that the that the wing

00:21:59.398 -- area S the larger the wing area,

00:22:02.320 -- the greater the lift you have.

00:22:05.290 -- OK, now this actually works well.

00:22:07.580 -- Not this relationship right here,

00:22:09.480 -- but this relationship.

00:22:10.953 -- If we wrote this is 1/2 Rho

00:22:14.496 -- v ^2 s think about a glider.

00:22:17.250 -- Moves very slow, right?

00:22:19.790 -- V is pretty small.

00:22:21.042 -- How does a glider make up for the

00:22:23.660 -- lift that it has to generate?

00:22:25.640 -- Was it have?

00:22:28.740 -- Lots of wing area.

00:22:30.344 -- Wings are very, very long,

00:22:32.350 -- so as if we wrote this as rho V squared.

00:22:36.080 -- Just substituting this right here.

00:22:37.940 -- If V is small then you could have

00:22:40.428 -- a large wing area right here to

00:22:43.223 -- generate whatever list you have nice.

00:22:45.770 -- OK, so that's the balance.

00:22:47.640 -- In aerodynamic design there OK?

00:22:50.770 -- We we've seen this diagram a few times and

00:22:53.920 -- showing the difference between a subsonic

00:22:56.374 -- airfoil you seem kind of nice rounded,

00:22:59.390 -- symmetric airfoil right there.

00:23:00.958 -- This is subsonic.

00:23:02.140 -- Speeds is getting close to transonic

00:23:04.408 -- here at the transonic regime.

00:23:06.450 -- Remember we said that now all the shockwaves

00:23:10.146 -- start to form right up there in the top.

00:23:13.870 -- And then that subsonic airfoil

00:23:15.900 -- has a bow shock here when it's

00:23:19.414 -- traveling supersonically,

00:23:20.810 -- whereas whereas a supersonic airfoil

00:23:23.765 -- like we see right here very thin.

00:23:27.830 -- OK, here you can see the shaded area right

00:23:30.827 -- here is going to give you it's roughly

00:23:33.616 -- related to the amount of drag that you see.

00:23:36.700 -- OK, so this shaded area right here

00:23:39.129 -- is smaller compared to its companion

00:23:41.223 -- there at the top same Mach number,

00:23:43.520 -- but different airfoil has

00:23:45.388 -- a nice rounded shape.

00:23:47.260 -- And then Supersonically right here,

00:23:48.920 -- you see, an oblique shockwave that forms.

00:23:51.870 -- Across that very thin airfoil,

00:23:53.290 -- whereas whereas you would have a bow shock

00:23:55.714 -- if you have a subsonic airfoil there.

00:23:58.250 -- OK, so again subsonic airfoils are great.

00:24:01.640 -- Flying subsonic Lee, they're awful.

00:24:04.060 -- Supersonically supersonic airfoils

00:24:05.512 -- are awful subsonic Lee,

00:24:07.450 -- but very good supersonically OK.

00:24:09.870 -- And then we talked a little bit

00:24:12.894 -- last time about these airfoils.

00:24:15.670 -- Here you see,

00:24:17.764 -- it's very thin.

00:24:19.860 -- K very thin right across here.

00:24:21.620 -- It's got a little curve there at the

00:24:23.764 -- bottom gives a little camber little

00:24:25.599 -- curvature so that you can generate lift.

00:24:27.800 -- Subsonic Lee.

00:24:28.722 -- This is the Russian.

00:24:30.570 -- This is the Russian aircraft that didn't

00:24:33.125 -- have that crashed really bad.

00:24:35.550 -- That's our 71 and then we talked about this

00:24:39.042 -- plane and F15 that flew with only one wing.

00:24:42.450 -- OK, Why was able to fly with one wing?

00:24:45.890 -- Well,

00:24:46.273 -- if we went back to our CISA Bell right?

00:24:49.720 -- He already went over it.

00:24:53.180 -- Right here, so it loses a wing.

00:24:55.880 -- So that means this area S decreases.

00:24:58.240 -- But as you increase the

00:24:59.920 -- Mach number right there,

00:25:01.270 -- you can generate the lift that you

00:25:03.552 -- need in order to fly in order to

00:25:06.088 -- in order to stay straight and level

00:25:08.430 -- roughly straight and level OK.

00:25:13.130 -- So actually before we get to that, here is.

00:25:19.100 -- So Mr Castle, sorry it's a Navy plane.

00:25:21.560 -- You OK with that? OK good.

00:25:24.310 -- So this is an F18 Hornet right here.

00:25:28.120 -- I want you to look. At the wings.

00:25:34.040 -- Can you get an idea for what that

00:25:36.152 -- for those wings look like right?

00:25:38.100 -- There? You see how thin those are?

00:25:40.870 -- It's got a very sharp leading edge.

00:25:44.130 -- OK, very thin all the way across there.

00:25:47.850 -- OK, now this actually has a really,

00:25:51.100 -- really cool design to fly.

00:25:53.430 -- Subsonic Lee. We saw F-14.

00:25:55.750 -- Tomcat had the swing wings.

00:25:58.080 -- Variable variable, swept wings.

00:25:59.860 -- If you look closely on

00:26:02.085 -- this airplane right here.

00:26:06.620 -- There is a hinge there and

00:26:09.608 -- the hinge right there. OK.

00:26:11.979 -- So what this plane does this is this

00:26:15.011 -- is awesome engineering as well.

00:26:17.960 -- OK, what this plane does with

00:26:20.234 -- this hinge here and the ailerons

00:26:22.601 -- here in the back. What it does?

00:26:28.420 -- OK, so as you say,

00:26:30.290 -- unhinged is probably a little bit unhinged,

00:26:32.900 -- so OK, so there's a hinge right there.

00:26:36.530 -- Flat plate for the surface of the wing

00:26:39.210 -- and then the other runs here in the back.

00:26:42.240 -- OK, so there's a kind of a classic

00:26:44.456 -- real thin supersonic airfoil.

00:26:46.270 -- What this leading edge hinge

00:26:47.965 -- does when it flies subsonic Lee.

00:26:49.970 -- Is that this actually bends down a little

00:26:52.906 -- bit and I'm just going to exaggerate

00:26:55.447 -- it so you can see what's going on.

00:26:58.450 -- So look like this.

00:27:01.320 -- Then the wing comes across here.

00:27:04.430 -- And then either on comes

00:27:05.855 -- back down there a little bit.

00:27:07.580 -- You see what that does to the wing?

00:27:11.290 -- Provides a little bit of curvature

00:27:14.170 -- little camber to the wing there so

00:27:17.438 -- that subsonic Lee you can take off and

00:27:21.124 -- land relatively relatively safely.

00:27:23.780 -- OK, that's that's pretty cool

00:27:25.880 -- design right there.

00:27:27.140 -- OK, will show some pictures a little

00:27:29.975 -- bit later on in the semester,

00:27:32.600 -- but one of the one of the research areas

00:27:35.750 -- that Aerodynamicists are working on now.

00:27:38.900 -- So let's say you're in the class,

00:27:41.840 -- which you really like materials.

00:27:43.940 -- OK, your control systems,

00:27:45.620 -- and not necessarily interested in that.

00:27:48.140 -- Fluids in the aero part they're

00:27:50.660 -- working water called smartwings.

00:27:52.340 -- OK, so smart wing.

00:27:55.400 -- So it has a surface like this cake

00:27:57.480 -- kind of a classic subsonic airfoil,

00:27:59.740 -- but this airport is actually

00:28:01.770 -- made up of a bunch of sections.

00:28:04.930 -- Right here. OK, on the wing.

00:28:08.774 -- So kind of looks like this,

00:28:11.390 -- and so those sections and

00:28:13.010 -- I'm just putting a squares.

00:28:14.630 -- I think they're hexagons.

00:28:15.926 -- I don't recall right off hand,

00:28:17.870 -- but you get the idea.

00:28:19.490 -- OK,

00:28:19.804 -- each of these little sections and

00:28:21.688 -- what they what they do depending on

00:28:24.008 -- the flight regime that you're in,

00:28:25.970 -- flying really fast or really slow,

00:28:27.920 -- it actually changes the shape

00:28:29.535 -- of the wing or your fly.

00:28:33.950 -- OK, so there are little.

00:28:37.460 -- Their little devices, little little motors,

00:28:39.530 -- little servos here on the inside that can

00:28:42.322 -- move up and down and change the shape of

00:28:45.297 -- the wing to make it the most aerodynamic

00:28:48.091 -- and the most the most efficient wing

00:28:51.070 -- for that particular flight regime.

00:28:54.050 -- That school engineering. OK.

00:28:59.040 -- Good, here's a here's another

00:29:01.895 -- little bit of supersonic design.

00:29:04.750 -- OK, I don't mean to.

00:29:06.570 -- I don't mean to brag on

00:29:08.682 -- this little guy right here.

00:29:10.570 -- Here's the X one.

00:29:11.966 -- We mentioned that it did break.

00:29:14.580 -- The sound barrier didn't have swept wings.

00:29:17.130 -- Not not the best compressible flow design.

00:29:19.670 -- Here's another part that Aerodynamicists

00:29:21.884 -- didn't understand at the time,

00:29:23.680 -- but is not a good design here is

00:29:27.192 -- that if you look at the tail.

00:29:30.490 -- OK, if you look at the tail right there,

00:29:33.700 -- notice it's got cut.

00:29:35.256 -- Classic tail design tail here and you see

00:29:38.286 -- you see here these tabs here in the back.

00:29:41.200 -- In the back there so that when this

00:29:43.816 -- is flying this remains flat and

00:29:45.979 -- then those back parts go up and

00:29:48.565 -- down again so that can maneuver

00:29:50.791 -- the plane going like this.

00:29:52.696 -- Turns out that Supersonically that is

00:29:55.054 -- not a very good way to maneuver in aircraft.

00:29:58.680 -- OK, so let's see why.

00:30:00.390 -- If this is flying super fast or if it's

00:30:03.261 -- flying this in the in the supersonic regime,

00:30:06.220 -- there were going to form there at the front.

00:30:12.230 -- Oblique shockwaves OK,

00:30:13.565 -- so an oblique shockwave forms there and

00:30:16.755 -- then these fins go up and down like this,

00:30:19.700 -- and so another shockwave is

00:30:21.665 -- going to format that side.

00:30:23.630 -- So you get 2 shocks that would form

00:30:26.014 -- and what aerodynamics is found is that

00:30:28.404 -- you could get a shock interaction

00:30:30.724 -- between those two shockwaves that

00:30:33.329 -- form their supersonic aircraft.

00:30:35.420 -- Nowadays will go back to

00:30:37.385 -- our F18 Hornet right here,

00:30:39.350 -- and you look at the tail.

00:30:42.720 -- Right here, OK, notice that there's no.

00:30:45.740 -- There's no part in the back there

00:30:49.408 -- that the whole tail moves up and down.

00:30:53.600 -- Like this or like this OK?

00:30:56.020 -- And it turns out that aerodynamically

00:30:58.438 -- again when this, when this turns,

00:31:01.363 -- say down like that.

00:31:03.650 -- Shockwave's going to form there at the

00:31:06.037 -- top and then you get smooth flow all

00:31:08.703 -- the way across the rest of the elevator.

00:31:11.420 -- OK, so that's another sign that you

00:31:13.583 -- could tell of what a supersonic

00:31:15.632 -- aircraft looks like if that whole

00:31:17.750 -- tail moves as opposed to just the

00:31:20.057 -- just the back portion of the tail.

00:31:24.300 -- Alright.

00:31:26.960 -- Aren't you gonna glad

00:31:27.896 -- you came to class today?

00:31:29.070 -- I'm glad that you came in class. OK.

00:31:33.690 -- Alaskan, so one of the things that now

00:31:36.554 -- that we've got this here, so let's see.

00:31:39.450 -- Did you send this to me?

00:31:41.610 -- Yes, so I got an email from Samantha.

00:31:44.490 -- He ran. She said that you should check

00:31:46.826 -- out this particular design and it's

00:31:49.187 -- called a coleopter Anna Coleoptile.

00:31:51.330 -- If you notice there that it has a,

00:31:54.210 -- it has a rounded wing on it.

00:31:57.880 -- That's that's kind of cool

00:31:59.755 -- right across there.

00:32:00.880 -- So here's the rounded wing.

00:32:02.760 -- Now it turns out that when

00:32:05.268 -- you have a finite wing.

00:32:07.520 -- Like you have here,

00:32:08.820 -- the pressure on the bottom of the

00:32:11.148 -- wing is higher than the pressure

00:32:13.014 -- on the top of the wing does.

00:32:15.150 -- That's how you generate lift.

00:32:16.740 -- But what happens is you have

00:32:18.504 -- what are called end effects.

00:32:20.240 -- So in end effect occurs when

00:32:22.052 -- the air on the bottom of the

00:32:24.311 -- wing kind of leaks out here,

00:32:26.280 -- and so you have high pressure air on the

00:32:29.133 -- bottom and low pressure air on the top.

00:32:31.690 -- And what that does is generate

00:32:33.826 -- these vertical structures go like

00:32:35.582 -- this and that causes a lot of drag.

00:32:37.730 -- It's a parasitic drag problem.

00:32:39.590 -- End effect drag problem OK in in

00:32:43.545 -- aerodynamics class you'd learn about how

00:32:46.926 -- wings can be shaped to minimize that.

00:32:50.610 -- The Spitfire that flew in World War

00:32:53.263 -- Two has elliptical shaped wings.

00:32:55.390 -- Beautiful beautiful aircraft but those

00:32:57.545 -- elliptical shapes right there minimized

00:32:59.759 -- the formation of those tip foresees.

00:33:01.750 -- OK so menace and minimizes

00:33:03.740 -- the drag flights faster.

00:33:05.340 -- OK so one of the nice things

00:33:08.028 -- about this design over here of

00:33:10.389 -- the coleopter is that it's got a

00:33:13.105 -- rounded wing so that there are no

00:33:16.024 -- tip effects that go across there.

00:33:18.489 -- Alright so you might think well.

00:33:20.910 -- How the heck does that generate

00:33:23.046 -- any kind of lift?

00:33:24.470 -- OK,

00:33:24.817 -- well it turns out that the angle

00:33:27.246 -- of attack is also related to the

00:33:29.756 -- amount of lift that you have,

00:33:31.950 -- so the higher angle of attack that you get,

00:33:35.150 -- the more lift you can generate.

00:33:37.290 -- Now for those that are paper airplane.

00:33:40.370 -- Enthusia STS

00:33:44.170 -- is it cold after I hear? Ever seen this?

00:33:48.130 -- And you can fly it. We'll see,

00:33:50.858 -- maybe we can catch this in slo-mo

00:33:53.266 -- from our from our from the cameras

00:33:55.583 -- here so you can fly this like this.

00:34:00.800 -- Not very good. Not very good.

00:34:04.260 -- But the idea is that you can generate lift.

00:34:07.500 -- Now you might think how can you

00:34:09.803 -- generate lift from something like that.

00:34:12.180 -- And again notice that in the picture

00:34:14.812 -- here it's flying at an angle of attack.

00:34:17.580 -- OK, so an airplane like this right here.

00:34:21.240 -- OK, actually has a slight angle of attack

00:34:24.008 -- built into it so that when it takes off.

00:34:26.710 -- In fact, if you if you look at it

00:34:29.374 -- straight and level right here, that Wing

00:34:31.812 -- has a little bit of an angle of attack.

00:34:34.760 -- OK, so that angle of attack is going to

00:34:37.460 -- create a small shockwave on the bottom.

00:34:39.920 -- Slight expansion wave on the top

00:34:41.852 -- so it generates lift.

00:34:43.140 -- OK, now it also answers the age old question

00:34:46.632 -- of how can a plane like this fly inverted?

00:34:50.330 -- OK, again, if this plane and you'll

00:34:52.283 -- notice it notice any aircraft when it

00:34:54.374 -- flies upside down screen level like

00:34:56.233 -- the Blue Angels and the Thunderbirds

00:34:57.871 -- when they do a flight show like that,

00:35:00.422 -- there's always a little bit of an

00:35:02.957 -- angle of attack as it goes down so

00:35:05.253 -- that it generates lift. OK, good.

00:35:10.130 -- That's your area lesson for today.

00:35:12.980 -- Alright. Got any questions?

00:35:14.924 -- Any questions? Anything we've seen so far?

00:35:18.570 -- OK.

00:35:21.500 -- So let's see here what we'd like to do now is

00:35:24.903 -- to figure out what we know for turning angle,

00:35:27.980 -- you can accelerate the flow.

00:35:29.600 -- What is the maximum turning angle?

00:35:31.540 -- How much can you turn?

00:35:33.810 -- And then what is the associated

00:35:35.670 -- Mach number associated with that?

00:35:37.230 -- OK, so here's our parental Meyer angle.

00:35:39.410 -- Theta of M is equal to this relationship,

00:35:41.900 -- right here it's got an arctangent of

00:35:43.972 -- the Mach number there and arctangent

00:35:45.881 -- of the Mach number right over here, OK.

00:35:48.439 -- So this is this will harken

00:35:50.353 -- back to your calculus days.

00:35:52.400 -- I know that was three or four years ago,

00:35:54.870 -- right so?

00:35:57.110 -- We're going to take a limit.

00:35:59.800 -- And what we're going to do is we're going

00:36:02.500 -- to take the limit as this angle as Phi.

00:36:05.330 -- We want to know what that

00:36:07.988 -- maximum turning angle is.

00:36:09.760 -- Of of our flow here.

00:36:11.500 -- So notice that in these arc

00:36:13.480 -- tangents right here we have gammas.

00:36:15.660 -- Cagamas are going to be constant for the air.

00:36:18.780 -- The only variable that we

00:36:20.515 -- have is the Mach number.

00:36:22.250 -- So we want to say if this

00:36:25.029 -- Mach number goes to Infinity.

00:36:27.460 -- What's the turning angle going to be?

00:36:30.510 -- OK, so just from mathematics right here.

00:36:33.560 -- The arctangent of Infinity is π / 2.

00:36:40.720 -- So we figure that one out.

00:36:43.170 -- Arctangent of Infinity is π / 2.

00:36:45.917 -- Let's go to the overhead here.

00:36:49.080 -- So have a triangle.

00:36:52.060 -- OK, and this is fi right here.

00:36:56.910 -- What's the tangent of Phi?

00:37:01.070 -- Say X. Why? Z what's tangent feet?

00:37:10.090 -- Tangent of Phi. The rise over the run. Y / X.

00:37:16.840 -- OK, alright, so let's let's

00:37:19.200 -- expand fi a little bit here.

00:37:26.040 -- Here is why an X.

00:37:30.280 -- 10s if he's going to get bigger

00:37:32.555 -- there right? 'cause your eyes.

00:37:34.314 -- Why is large and X is small here?

00:37:37.390 -- What happens when you have

00:37:38.560 -- a triangle look like this?

00:37:46.200 -- Look at this rise divided by

00:37:48.858 -- that teeny tiny run right there.

00:37:51.840 -- And then eventually.

00:37:53.553 -- You just have a straight line.

00:37:56.980 -- This is 90 degrees right here.

00:37:59.530 -- Which is the same. Is π / 2?

00:38:05.520 -- So the arctangent.

00:38:07.371 -- A fee from our little trigonometry

00:38:11.073 -- bit right there is π / 2.

00:38:14.287 -- That's why the limit as

00:38:16.972 -- fee tends to Infinity. OK.

00:38:21.280 -- Net fee is just the rise over the run.

00:38:23.900 -- That's why are y / X here is equal to π / 2.

00:38:27.653 -- OK, so if we take that limit then

00:38:29.757 -- Theta taking the limit here as

00:38:31.596 -- N goes to Infinity here an as N

00:38:34.053 -- goes to Infinity here substituting

00:38:35.528 -- in π / 2 to both of those terms,

00:38:38.160 -- we get that Theta is equal to π /

00:38:40.780 -- 2 times the square root of gamma

00:38:42.803 -- plus one divided by gamma minus one.

00:38:44.850 -- All this minus one right here.

00:38:47.720 -- So you can actually get a turning

00:38:51.066 -- angle of 130 degrees.

00:38:53.440 -- If you could turn that flow 130 degrees,

00:38:55.910 -- you would accelerate it to Infinity.

00:38:59.920 -- Obviously that's not going to happen OK,

00:39:02.030 -- Anna 130 degrees.

00:39:02.930 -- I mean, if you think about it,

00:39:05.040 -- it's got not just 90 degrees,

00:39:06.840 -- but now it's coming back

00:39:09.105 -- in the opposite direction.

00:39:10.920 -- Not an actual physical limit,

00:39:12.320 -- but a theoretical limit on what

00:39:14.474 -- the term could be in your flow.

00:39:16.940 -- OK, little bit of mathematical

00:39:20.750 -- gymnastics there OK?

00:39:23.040 -- This is the problem that was

00:39:24.912 -- cancelled in your homework assignment,

00:39:26.850 -- but I want you to make sure that

00:39:29.258 -- we've got shock tubes down so that you

00:39:32.133 -- understand how a shock tube works.

00:39:34.460 -- OK, so again,

00:39:35.264 -- this is 1/2 of it as a rehash from

00:39:38.039 -- when we did normal shocks and moving

00:39:40.909 -- normal shocks in a shock tube.

00:39:43.110 -- You have a driver section.

00:39:44.840 -- OK, high pressure gas right?

00:39:46.570 -- Over here there's a membrane that

00:39:48.856 -- separates this high pressure region

00:39:50.786 -- from a low pressure region right over here.

00:39:53.520 -- And this is what the pressure

00:39:55.986 -- profile looks like.

00:39:57.220 -- So you have a high pressure

00:39:59.542 -- region in the driver region.

00:40:01.740 -- A low pressure area right

00:40:03.795 -- here in the DRIVIN region.

00:40:05.850 -- OK then we break the membrane

00:40:08.406 -- and what that membrane does is

00:40:10.967 -- that it creates a normal shock

00:40:13.277 -- and that normal shockwave then

00:40:15.692 -- propagates down from left to right.

00:40:18.810 -- OK,

00:40:19.214 -- and so this stuff that we talked

00:40:22.042 -- about in class when we did

00:40:24.901 -- normal shops right across here.

00:40:27.350 -- Right across here,

00:40:28.307 -- we figured out what the pressure

00:40:30.602 -- profiles the Mach number is.

00:40:32.200 -- The Mach number of the wave, and so on.

00:40:35.225 -- We figured all this stuff out and we

00:40:38.262 -- ignored the things on the left hand side OK.

00:40:41.780 -- Kate, now that we know about expansion waves.

00:40:44.930 -- What's going on here between 3:00 and

00:40:47.450 -- 1:00 is an expansion wave problem.

00:40:50.330 -- Just like we talked about before.

00:40:53.610 -- OK,

00:40:54.067 -- so what's happening here is that again

00:40:57.266 -- you have a high pressure region here.

00:41:00.780 -- There is a lower pressure region

00:41:02.898 -- right across here in Region 3.

00:41:04.980 -- In fact,

00:41:05.714 -- if you look at the pressure

00:41:07.916 -- profile here it is.

00:41:09.180 -- This is the driven section

00:41:10.930 -- has a low pressure,

00:41:12.330 -- goes across the shock.

00:41:13.610 -- That's what this region is right

00:41:15.597 -- across here and then that pressure

00:41:17.727 -- remains constant to what's called

00:41:19.582 -- the contact surface right here.

00:41:21.430 -- And that's where Region 3 contacts Region 4.

00:41:24.230 -- And then there is a gradual

00:41:26.606 -- increase in the pressure right up

00:41:29.075 -- here and then the driver section.

00:41:31.480 -- OK,

00:41:32.021 -- so this gas this high pressure gas

00:41:35.808 -- begins to expand. As it goes across.

00:41:40.872 -- OK.

00:41:41.640 -- Draw that out, drawn out process out here,

00:41:44.900 -- OK?

00:41:46.430 -- So.

00:41:52.870 -- So here's the membrane.

00:41:54.442 -- Here's the high pressure section.

00:41:56.410 -- Here is the low pressure section here.

00:41:59.160 -- OK, when the membrane breaks.

00:42:03.680 -- Creates a shockwave.

00:42:06.900 -- Propagates down this direction here,

00:42:09.680 -- and an expansion wave

00:42:12.464 -- forms as this gas expands.

00:42:15.950 -- Into this region right over

00:42:18.695 -- here so that wave propagates.

00:42:21.440 -- Starts out small here,

00:42:23.416 -- gets a little bit larger.

00:42:25.890 -- And then that distance increases

00:42:27.525 -- right across there as this high

00:42:29.536 -- pressure gas region expands

00:42:30.820 -- and goes across this way.

00:42:35.040 -- As a as my kids say, you know fun

00:42:39.299 -- fact fun fact about expansion waves.

00:42:42.140 -- OK, this behaves exactly like.

00:42:46.240 -- A traffic jam or traffic

00:42:49.620 -- flow at a red light. OK.

00:42:54.940 -- But how could that be OK?

00:42:57.100 -- Think about this.

00:42:58.180 -- You have a red light right here.

00:43:04.440 -- We want to make this authentic,

00:43:06.310 -- so there's a red light right there.

00:43:08.480 -- OK? And there are cars lined up.

00:43:11.840 -- And it's all bumper to bumper, right?

00:43:13.897 -- I'm sure all of you keep a safe

00:43:16.593 -- distance right when you break.

00:43:18.520 -- In traffic, especially,

00:43:19.669 -- the traffic jams here in Moscow ID OK,

00:43:22.620 -- so looks like this and you can have a line

00:43:26.893 -- of cars that go all the way back here.

00:43:30.670 -- OK, so the red light then turns green.

00:43:35.490 -- Green light right here.

00:43:40.130 -- There's a green light OK,

00:43:42.170 -- and now the cars go do all seven of

00:43:45.050 -- these cars move at the same speed and

00:43:48.453 -- propagate through their drive-thru light?

00:43:51.150 -- No, what happens?

00:43:52.809 -- First car goes it's here.

00:43:55.580 -- And there's a larger distance

00:43:57.390 -- between that one and the next one,

00:43:59.880 -- and then that distance gets a

00:44:01.890 -- little bit smaller and smaller,

00:44:03.810 -- and the way in the back here is that,

00:44:07.040 -- and I've had this happen to me before.

00:44:09.900 -- Is that you see the green light here,

00:44:12.760 -- but the distance when I'm parked

00:44:14.710 -- way back here the distance between

00:44:16.863 -- me and that car hasn't changed.

00:44:19.210 -- Heckuva lot for 1015 seconds,

00:44:21.000 -- however long it takes to

00:44:22.790 -- propagate that through these cars.

00:44:24.580 -- Driving through here.

00:44:26.062 -- It turns out have the same

00:44:29.026 -- mathematical modeling associated

00:44:30.809 -- with expansion ways right there.

00:44:33.820 -- Same modeling process here.

00:44:35.308 -- This membrane breaks the gases expand.

00:44:37.540 -- This wave goes faster than this one,

00:44:40.140 -- which goes faster than this one and so on.

00:44:43.490 -- So as this as this pressure wave

00:44:46.381 -- propagates there through the back

00:44:48.431 -- has the same properties as cars

00:44:50.645 -- parked not part but in a in a

00:44:53.369 -- traffic line right across there.

00:44:57.160 -- There you can tell your folks

00:44:58.984 -- over the break that you learned

00:45:01.030 -- something here in gas dynamics.

00:45:02.940 -- OK, like I said, fun fact.

00:45:05.720 -- OK, shock tubes in let's go and here is

00:45:09.374 -- what happens after the membrane breaks.

00:45:13.010 -- OK, so let's look at the

00:45:15.920 -- expansion wave process.

00:45:17.380 -- OK, so here again these waves

00:45:19.840 -- propagate back and as they propagate

00:45:22.606 -- back eventually this high pressure

00:45:25.216 -- region is going to completely expand.

00:45:28.560 -- OK, so the initial state of our

00:45:32.179 -- of our shock tube right here.

00:45:35.730 -- Of this pressure difference

00:45:36.918 -- across here from one to two,

00:45:38.700 -- we have a high pressure region

00:45:40.380 -- in a low pressure region,

00:45:41.970 -- right across here,

00:45:42.858 -- that's called the diaphragm pressure ratio.

00:45:46.930 -- OK, right across there after that

00:45:49.348 -- diaphragm or the membrane breaks,

00:45:51.510 -- then the pressure in Region 3 is the

00:45:55.190 -- same as the pressure in region 4.

00:45:58.840 -- And the speed in Region 3 is the

00:46:01.216 -- same as the speed in region 4.

00:46:03.520 -- So if we go back to our

00:46:05.627 -- little drawing right here,

00:46:06.950 -- these two pressures are the same.

00:46:08.820 -- Cross this contact surface and the speed

00:46:11.004 -- of the air or the gases between those

00:46:13.570 -- two regions are the same as well. What?

00:46:16.176 -- What do you think would be different?

00:46:19.580 -- Speeds the same pressures the same.

00:46:21.010 -- What do you think would be

00:46:23.380 -- different across there?

00:46:24.570 -- What happens to the temperature

00:46:27.760 -- across a shockwave?

00:46:29.680 -- It goes up what happens to the

00:46:32.214 -- pressure downstream of this expansion

00:46:34.112 -- wave as it starts to expand.

00:46:36.320 -- Goes down. OK, so there's a.

00:46:38.530 -- There's a temperature difference between

00:46:40.365 -- 3:00 and 4:00, right across there.

00:46:42.586 -- OK, T4 is going to be higher than T2,

00:46:45.890 -- and T3 is going to be higher than T1.

00:46:49.830 -- That's really what defines that difference

00:46:51.882 -- right across there is that temperature OK,

00:46:54.380 -- but the pressures in those

00:46:56.130 -- two regions are the same.

00:46:57.880 -- The speed in those two regions of the same.

00:47:01.030 -- OK, so again, across an expansion wave,

00:47:03.480 -- the flow is isentropic.

00:47:04.764 -- So we can use our isentropic relations here.

00:47:07.680 -- P3 or P1 is equal to T3 over T1 to

00:47:10.572 -- the gamma over gamma minus one power.

00:47:13.630 -- Or we could write that in terms

00:47:16.843 -- of the densities as well.

00:47:19.090 -- You know this is nice because

00:47:20.590 -- once we get it in this form,

00:47:22.390 -- we can write this in terms

00:47:24.412 -- of the speed of sound.

00:47:26.330 -- So now P3 over P11 minus gamma one.

00:47:30.120 -- Over 2 times V 2 /, 81 squared.

00:47:33.334 -- So this is the Mach number.

00:47:36.680 -- OK, in the expansion region right there to

00:47:39.272 -- the two gamma sub one over gamma minus one,

00:47:42.260 -- right across there.

00:47:43.769 -- OK if we solve for V then.

00:47:47.290 -- We can solve for V2.

00:47:48.760 -- We can solve for the speed and region 2.

00:47:52.260 -- Right across there and it is just

00:47:54.808 -- a function of the speed of sound.

00:47:57.450 -- He said one and this pressure ratio,

00:48:00.050 -- the diaphragm pressure ratio P2 over P1 OK.

00:48:04.450 -- What it gives us here is that since gamma

00:48:07.105 -- minus one this power right over here,

00:48:09.410 -- gamma minus 1 / 2 gamma is less

00:48:11.658 -- than one as PP1 tends to Infinity.

00:48:14.060 -- This term is going to go to zero,

00:48:16.540 -- and So what this does is you

00:48:18.990 -- can generate a maximum.

00:48:20.780 -- Speed in region 2.

00:48:22.528 -- From this term right there twice,

00:48:25.150 -- the speed of sound divided

00:48:26.930 -- by gamma minus one.

00:48:28.360 -- OK, enough gammas for air,

00:48:29.850 -- that's one point 4 -- 1 or that's two

00:48:32.588 -- times the speed of sound divided by .4.

00:48:34.920 -- What it tells you can do is you

00:48:37.608 -- can generate pretty high speeds.

00:48:39.820 -- With the shock tube.

00:48:42.040 -- OK, they don't last for very long.

00:48:44.230 -- OK,

00:48:44.522 -- but you can get hypersonic flows from a

00:48:46.858 -- shock tube that comes right across here,

00:48:49.240 -- OK?

00:48:51.270 -- Good,

00:48:51.614 -- so that was the problem

00:48:53.334 -- that was cancelled is

00:48:54.781 -- to figure out the expansion portions

00:48:56.983 -- of the flow. OK, any questions? Yes.

00:49:04.810 -- OK, I'm getting to that,

00:49:06.520 -- so I answer part one.

00:49:08.220 -- No shock tube questions,

00:49:09.572 -- no moving shock problems on the exam.

00:49:14.180 -- What's that? And an airfoil

00:49:18.041 -- is not a shock tube, so yes,

00:49:20.052 -- so there is an airfoil problem.

00:49:21.770 -- There is not a shock to problem.

00:49:23.780 -- OK, I just want to make sure that

00:49:26.068 -- you understood kind of the basic

00:49:27.825 -- workings of what a shock tube are.

00:49:29.810 -- OK, not answer your question here.

00:49:31.530 -- Exams are open book.

00:49:33.690 -- Open notes OK.

00:49:37.600 -- And it's also open laptops if you

00:49:40.414 -- choose that you've got a laptop to use

00:49:43.203 -- com prop that is OK to use as well,

00:49:45.950 -- but there are no communications

00:49:48.122 -- with the outside world or the

00:49:50.278 -- inside world on your laptop.

00:49:51.870 -- So if you have a phone 'cause you

00:49:54.558 -- want to use your app, that's OK too.

00:49:59.430 -- OK, so open book, open notes, open com,

00:50:02.910 -- prop, or if you've got some other,

00:50:05.960 -- you know if you use some other device

00:50:08.848 -- for your compressible flow tables,

00:50:11.610 -- that's OK as well, yes?

00:50:17.110 -- Open Book open notes.

00:50:24.070 -- There are five problems on the exam,

00:50:25.450 -- so if you spend all your time looking

00:50:26.946 -- at your notes, you're not going to

00:50:28.408 -- have time to solve the problems.

00:50:32.510 -- OK, so be sure that your

00:50:34.682 -- your laptops are charged.

00:50:36.130 -- We don't have a whole lot of.

00:50:38.660 -- Let's see. Do you have?

00:50:40.470 -- Do you have plugs underneath?

00:50:44.200 -- Not sure what you got there. OK.

00:50:48.770 -- And you will have the hour and 15

00:50:51.746 -- minutes to solve the exam as well.

00:50:54.520 -- OK, I will talk to the greater today and we

00:50:57.548 -- will try and get this graded by tomorrow.

00:51:00.370 -- So if you want to pick him up

00:51:02.818 -- tomorrow you could come by my

00:51:04.854 -- office and pick up the exams.

00:51:06.870 -- That's fine.

00:51:07.616 -- The solutions are also will

00:51:09.481 -- be available this afternoon.

00:51:11.310 -- So even though even if you don't

00:51:12.689 -- have your homework assignment,

00:51:13.750 -- if the greater can't get it back in time.

00:51:17.120 -- Can you can always look online on TV learn?

00:51:21.160 -- OK. Good, so let's see here.

00:51:24.980 -- I would say that the exam problems

00:51:26.807 -- are going to be very similar to what

00:51:28.959 -- you see in the homework problems.

00:51:30.830 -- Let's just let's just review.

00:51:33.610 -- Write reviews 'cause then you

00:51:35.720 -- can see how intelligent you have

00:51:38.277 -- become over the last one week.

00:51:40.500 -- Six now, let's see what we've discussed here,

00:51:43.740 -- OK?

00:51:45.680 -- Just going through just going through the

00:51:49.460 -- going through the table of contents here.

00:51:53.280 -- OK, let's see here.

00:51:54.708 -- You probably won't have any problem that

00:51:57.396 -- just discusses the continuity equation.

00:51:59.780 -- You probably won't have a problem

00:52:02.222 -- that just has the ideal gas law,

00:52:05.050 -- but you could certainly use it.

00:52:08.650 -- OK,

00:52:09.065 -- some fundamental aspects

00:52:10.310 -- of compressible flow.

00:52:11.560 -- You know how to calculate the speed of sound?

00:52:16.350 -- OK, and you know how to calculate

00:52:18.380 -- Mach numbers and Mach waves.

00:52:19.950 -- I would say that's going to

00:52:21.840 -- be something to study up on.

00:52:23.550 -- Make sure that you understand what

00:52:25.224 -- a Mach wave is and how different

00:52:27.348 -- it is from an oblique shockwave.

00:52:29.250 -- What is the difference?

00:52:32.300 -- What's the difference between an

00:52:33.910 -- oblique shockwave animac wave?

00:52:37.280 -- A Mach wave is an infinitesimally

00:52:40.880 -- weak oblique shockwave.

00:52:42.680 -- OK. Good. Uh, now?

00:52:49.560 -- You probably won't have a problem.

00:52:50.820 -- It says, just calculate the speed of sound,

00:52:52.500 -- but you will likely have a problem.

00:52:53.970 -- You'll have to calculate the speed of sound.

00:52:56.440 -- OK, so this is all fundamental

00:52:59.068 -- stuff right here.

00:53:00.390 -- OK, in chapter four we learned about

00:53:02.938 -- isentropic flows and the difference

00:53:05.251 -- between stagnation conditions,

00:53:06.980 -- static conditions and critical

00:53:08.732 -- conditions right there, OK?

00:53:12.020 -- You probably won't have a problem

00:53:14.138 -- that says what is the stagnation

00:53:16.500 -- temperature of such and such.

00:53:18.680 -- You might OK, but you'll be able to.

00:53:21.820 -- You'll need to calculate what stagnation

00:53:25.144 -- temperatures are and pressures.

00:53:27.360 -- Pretty straightforward using

00:53:28.542 -- the tables using comp.

00:53:30.120 -- However you want to do it OK.

00:53:32.880 -- We learned about shockwaves.

00:53:35.072 -- OK, and you'll probably see some problems

00:53:38.584 -- that have a normal shocks in him.

00:53:41.890 -- OK, that's very important in gas dynamics.

00:53:46.210 -- OK, pitot tubes. Are also important,

00:53:51.190 -- so make sure you got those down.

00:53:54.150 -- Don't worry bout moving normal shocks.

00:53:58.200 -- OK.

00:54:00.360 -- Are oblique shock waves are important?

00:54:02.620 -- Make sure that you've got those down

00:54:05.147 -- and expansion waves are also important.

00:54:07.520 -- Make sure that you've got that down.

00:54:12.350 -- And make sure that you've got

00:54:14.702 -- the reflection part of oblique

00:54:16.895 -- shockwaves down pretty well. OK.

00:54:21.310 -- Again, as you learn in your problems,

00:54:23.570 -- a lot of it's just the geometry.

00:54:25.830 -- Make sure you've got the

00:54:27.890 -- direction of the flow right. OK.

00:54:32.670 -- Just to re clarify. We're going to

00:54:36.557 -- go back to our normal shocks here.

00:54:42.040 -- Region 1. Region 2 flow goes

00:54:44.950 -- this way comes out this way.

00:54:47.930 -- This is the region that is upstream.

00:54:53.590 -- It is also ahead. Of the shock.

00:54:59.930 -- OK, this is the supersonic flow.

00:55:03.280 -- Regime this is the subsonic flow.

00:55:07.240 -- Regime this is. Downstream.

00:55:12.700 -- Of the shock, this is also behind.

00:55:17.250 -- The shock there means the same thing.

00:55:21.530 -- OK, sometimes a problem will ask

00:55:23.396 -- what's the temperature and

00:55:25.004 -- pressure just downstream of the shock?

00:55:26.870 -- Where does that where you where?

00:55:28.750 -- Are you looking to calculate that?

00:55:31.280 -- Right there.

00:55:34.070 -- Just downstream so you don't have to

00:55:35.876 -- worry about any of the flow anywhere else.

00:55:38.000 -- What's the temperature and pressure?

00:55:39.310 -- The Mach number just downstream of the

00:55:41.564 -- shock just in that region right there?

00:55:44.080 -- OK. Excellent. Any questions?

00:55:52.850 -- OK, have a wonderful day study

00:55:55.724 -- up for the exam and we'll see

00:55:59.529 -- Thursday bright and early.

Duration:"00:49:15.0530000"



00:00:27.640 -- Alright, for today we're going to start in Chapter 3. We're

00:00:31.336 -- going to go over. Some were basically kind of reviewing at

00:00:35.032 -- this point, so a couple of things to show everybody on the

00:00:39.064 -- website. Is if we go to I have to move this up here. Sorry I

00:00:45.806 -- forgot I have a preview in a program one so up here the

00:00:50.928 -- lectures we have. Intro class review so for this I have been

00:00:55.656 -- an I will continue to do so. Uploading my intro class of

00:01:00.384 -- lectures. Hopefully most of these links should work OK good

00:01:04.324 -- and they were working on the right files. That's even

00:01:08.264 -- important. Super important

00:01:09.446 -- actually. So there's the.

00:01:13.100 -- And.

00:01:14.470 -- Basic Intro class review lectures. Like I said, I'll

00:01:16.882 -- be putting up a whole bunch more, especially as we run

00:01:19.830 -- into more stuff that's more pertinent to the stuff we're

00:01:22.510 -- looking at and or reviewing.

00:01:25.260 -- And on today's we are going to be in Module 3 which just from

00:01:28.550 -- correspond to chapter three. I'm not quite sure why use the term

00:01:31.370 -- module, but I did and there it is, so we're using it.

00:01:35.560 -- So here my links are working. Yeah, I have a few more links

00:01:39.408 -- for other things to look at, so my central limit theorem we're

00:01:42.960 -- going to talk about that today review that I have two

00:01:46.216 -- different lectures for that. One of 'em actually shows a

00:01:49.176 -- simulation which will be. I don't know. I always enjoyed

00:01:52.136 -- this simulation. Once I finally thought so it was kind of nice

00:01:55.688 -- to see. And we're also going to go through this probability

00:01:58.944 -- distributions handout. I wasn't actually going to put this up

00:02:01.904 -- and I'm going to do most of it on the document camera, but.

00:02:07.070 -- I decided to at the last minute and it literally is a last

00:02:10.567 -- minute hand out, so don't expect anything gorgeous. No pretty

00:02:13.257 -- colors, sorry, no pretty colors here. Have a couple of nice

00:02:16.216 -- looking tables or just not sitting where I want them to,

00:02:19.175 -- but the handout itself will work just fine, and that's actually

00:02:22.134 -- primarily we're going to go through today and then we'll see

00:02:25.093 -- if we get a chance to look at least one of these central limit

00:02:29.128 -- theorem. Handouts, so today we are going to look at this, but

00:02:33.546 -- we are going to walk through all this, so we want to do most of

00:02:37.836 -- this on the document camera, but I wanted to show something

00:02:40.982 -- first, because, well, this thing can graph so much nicer than I

00:02:44.414 -- can. So alright first.

00:02:49.140 -- We need to review some basic terms from

00:02:53.228 -- probability and we want to.

00:02:57.150 -- Zoom in just to hear will come back to the computer in just a

00:03:02.358 -- bit. So remember, we're going to be talking about is

00:03:06.078 -- probability distributions.

00:03:10.950 -- We'll start out with this simple case just to work through. Now

00:03:14.826 -- that we're going to get into a super highly complex one, but

00:03:18.702 -- will start out with a simple case. Alright, so we have this

00:03:22.578 -- hypothesize data, and that's what this worksheet that I made

00:03:25.808 -- is going to basically following through it. Oh, sorry

00:03:28.715 -- hypothesis. Can you tell that to normal term? I use hypothesized.

00:03:36.740 -- Population.

00:03:41.420 -- And it's an old example. It's probably not extremely current

00:03:45.010 -- in terms of its probabilities. Fitting it is OK. It will still

00:03:49.318 -- work. So here we have the number of TV sets that are owned.

00:03:55.960 -- Per household.

00:04:01.570 -- Nowadays it might be more interesting to look at

00:04:05.960 -- phones or computers, but everybody's got something

00:04:09.033 -- alright. So in this population, well, we can

00:04:12.545 -- either have the TV's can take on values of 0123 or four. Do

00:04:18.252 -- I have for USF 4?

00:04:21.920 -- And then we have some probabilities associated

00:04:23.845 -- with those.

00:04:26.400 -- P of TV's.

00:04:29.250 -- So the probability of those.

00:04:31.860 -- So I'm just going to write

00:04:33.006 -- another wreath. We can make a nice pretty table here

00:04:35.404 -- when we're done.

00:04:40.190 -- Alright.

00:04:42.870 -- OK, so this example is at least.

00:04:46.220 -- Now might be over 10 years old, but it's at least 10 years old.

00:04:50.100 -- So the probability that probably not quite so accurate anymore,

00:04:53.140 -- but that's OK for what we want to do here. So this is the

00:04:57.396 -- number of TV sets owned per household. And if you want to

00:05:01.044 -- think about it this way for remember what this term is

00:05:04.692 -- the number of TV's this is going to be a random variable.

00:05:10.630 -- Which remember is kind of like a function of valued function.

00:05:15.910 -- So a specific value of our distribution has a specific

00:05:19.610 -- probability associated with it.

00:05:23.440 -- Alright.

00:05:25.470 -- So actually, let's make a nice table. I should have done that

00:05:28.350 -- to begin with, but whatever.

00:05:30.770 -- I do things the hard way sometimes, so TV's

00:05:34.991 -- probability of TV's.

00:05:43.660 -- Your attentive and you want to redo it, go

00:05:45.928 -- for it, I understand.

00:05:48.120 -- Probably not necessary, as long as you got all your information,

00:05:51.442 -- but nice little table.

00:05:54.160 -- I could have done it vertically, whatever it however you want to

00:05:57.280 -- look at it. Either way, this will get us the basic idea.

00:06:02.790 -- And that's where my graph comes into play and I totally draw it.

00:06:06.287 -- But really, my little – my little handout can show so much

00:06:09.515 -- better than I could ever draw it. So if you want to look at

00:06:13.281 -- that real quick on the computer that is distribution TV's.

00:06:17.910 -- Thought about playing with colors, but I just left alone.

00:06:20.390 -- Figured you could get the gist of it. So we got our 20% here at

00:06:24.110 -- zero and two 40% at one and then our 10% at three and four.

00:06:28.490 -- Alright. So back to our examples here.

00:06:35.110 -- So one of the things, well, we have many things of interest

00:06:38.254 -- that we'd like to look at about. One of the major things

00:06:41.398 -- we want to look at is to look at some of our summary

00:06:44.804 -- statistics, and while looking at this, it would probably be

00:06:47.424 -- nice to know on average, how many TV's are owned per

00:06:50.306 -- household. So we want to find a mean.

00:06:53.970 -- And we also call this here in. With this we call this

00:06:58.134 -- an expected value.

00:07:02.070 -- Expected value alright, so an expected value is a mean, but

00:07:06.756 -- unlike our continuous distributions versus discrete

00:07:09.312 -- and this is more of one of those discrete answers 'cause you

00:07:14.424 -- can't own. 1 1/2 television sets. You could on average but

00:07:19.110 -- not. Literally.

00:07:22.380 -- You probably don't want broken ones. I think they think they're

00:07:26.208 -- counting functional TV's not nonfunctional TV's as well, so

00:07:29.340 -- this would be discreet.

00:07:33.200 -- So basically you want to think about it that your variable

00:07:36.344 -- in our book uses why a lot versus X, but pick a letter. It

00:07:40.012 -- doesn't really matter. I'm going to use why just because their

00:07:42.894 -- book does, but what was I going with? This whole number values?

00:07:46.860 -- That's what these things can take on.

00:07:50.850 -- There's an S there. There we go.

00:07:54.640 -- So this mean here the way we're going to compute it is because

00:07:59.359 -- it's basically it's a weighted average, so not every value of

00:08:03.352 -- our random variable TV's.

00:08:05.560 -- Takes on equal probabilities, they don't have equal

00:08:08.240 -- probabilities, so we have a weighted average that we're

00:08:11.255 -- going to do.

00:08:13.440 -- And.

00:08:15.690 -- Since we're dealing with the population, this is what we

00:08:18.900 -- call deductive because we know exactly what's going to

00:08:21.789 -- be happening in the population versus a sample,

00:08:24.357 -- and most of our exploration there is going to be

00:08:27.567 -- inductive, but this ones deductive, because we can

00:08:30.135 -- actually see what's actually happening, so we're going to

00:08:33.024 -- call this thing mu the population mean of the

00:08:35.913 -- distribution, some other notation E of Y, like

00:08:38.481 -- function notation.

00:08:41.180 -- And to calculate this is the

00:08:44.180 -- sum. So Sigma sum at each Y times its

00:08:50.164 -- corresponding probability.

00:08:53.320 -- They just calculate about products and add them all up.

00:08:59.830 -- Alright, so why not? We're here. We should do this, for example.

00:09:05.870 -- So to calculate our expected value of Y or mean for this

00:09:11.258 -- we would take zero times its probability.

00:09:15.950 -- Plus one, so I was trying to parenthese ahead of time times

00:09:19.922 -- the probability of 1.

00:09:22.890 -- Two times its probability plus three times .1.

00:09:30.060 -- There's four times by 1.

00:09:33.490 -- There's also and then we get a lovely 1.5.

00:09:38.590 -- Oops, sorry, papers got broken.

00:09:42.410 -- So on average, we could expect a household to

00:09:45.830 -- have about 1 1/2 TV's.

00:09:49.290 -- It's like the 1 1/2 kids thing, though obviously we can't have a

00:09:51.994 -- half a TV or a half a kid, but it's an average, even if it's

00:09:55.114 -- not a part of the original

00:09:56.362 -- distribution. And that's OK.

00:09:59.780 -- So. This, unfortunately, you're going to have to torture with my

00:10:03.740 -- drawing anyways. If I drew out our little distribution.

00:10:09.860 -- So this will be our probability.

00:10:12.750 -- We have wide on the X axis.

00:10:16.310 -- Let's see here.

00:10:18.840 -- So I'm just kind of guesstimating I'm not an artist

00:10:22.290 -- by any stretch of the imagination. I can draw a decent

00:10:26.085 -- Bell curve. And occasionally decent rectangles.

00:10:31.040 -- Pretend those are both .1 and the other ones are point 2.4,

00:10:35.348 -- point 2.1. .1 there we go.

00:10:40.090 -- So if we imagine where we put the mean, that would be right

00:10:43.639 -- about here. Well, this is basically what we would consider

00:10:46.369 -- this center of mass. So if we actually try to balance this

00:10:49.645 -- thing on, that's exactly the point where it would balance the

00:10:52.648 -- center of mass right there. And that's where that mean is.

00:10:59.580 -- I think I just wanted to touch you with my drawing.

00:11:01.395 -- That's not what I think I needed to do here.

00:11:04.750 -- Right, and of course we love measures of location. That's

00:11:08.310 -- what the mean is. But we also love measures of spread so we

00:11:12.938 -- can see how much variation we actually have. So this is our

00:11:17.210 -- measure.

00:11:19.070 -- Of spread.

00:11:21.380 -- Variation.

00:11:27.190 -- It's one of 'em, but this one in particular.

00:11:31.040 -- The variance.

00:11:34.270 -- Is the average.

00:11:40.050 -- Important work here squared.

00:11:43.820 -- Distance.

00:11:46.410 -- Each point is from its mean.

00:11:53.280 -- Remove there.

00:11:59.540 -- So we can see how much variation we have in our data. Points were

00:12:03.670 -- in relation to the center.

00:12:05.750 -- Of the distribution.

00:12:11.170 -- At see here it's units.

00:12:14.290 -- R-squared units.

00:12:22.720 -- Measurement but yeah.

00:12:25.130 -- But it's not on the same scale as the mean, so not all the

00:12:29.568 -- time. Is this the one we want to directly deal with? But we still

00:12:34.006 -- have to calculate it. So to do that it's it's Greek symbol is a

00:12:38.444 -- Sigma squared. Yeah, my Sigma is mostly OK.

00:12:42.810 -- And one of my friends used to draw it and it looked like a

00:12:44.896 -- Theta and I was like Theta

00:12:45.790 -- squared shoes. Now it's a Sigma. OK, mine supposed to be a Sigma

00:12:50.030 -- at mostly kind of looks like one. You can also use via Y. Now

00:12:54.930 -- this V here is going to denote the actual true variance.

00:12:59.660 -- And of course, since we're dealing with the population,

00:13:01.694 -- that's OK. 'cause that's what we're going to be looking at.

00:13:04.180 -- But that reason I brought that up is that will come into play

00:13:07.118 -- here in just a bit, so.

00:13:09.070 -- Keep that in the back of your

00:13:10.183 -- head, all right. So what we're going to do

00:13:14.456 -- is look at Y minus mu.

00:13:18.550 -- Quantity squared times the probability of Y, so we'll take

00:13:23.300 -- each squared difference of each value between it and the mean.

00:13:29.170 -- Look at that squared distance and multiply it by the

00:13:32.280 -- probability of that data point and that gives us.

00:13:35.970 -- What we're looking for in terms of the variation.

00:13:42.290 -- Alright. So of course we're going to do that.

00:13:48.350 -- And I got a little carried away on my hand out, which is OK and

00:13:52.910 -- carried away in a good way, sort of. It might be a little

00:13:56.862 -- redundant for you, but I did actually expand some of these

00:14:00.206 -- formulas a little bit more, but I did show the actual work later

00:14:04.158 -- on, so we will actually do this. So Sigma squared is the variance

00:14:08.110 -- of Y. So we're going to take the first data point, which is a 0.

00:14:13.780 -- Minus the mean.

00:14:15.750 -- Squared and the probability of zero was a .2.

00:14:21.720 -- And we get to do this for all

00:14:23.704 -- five values. So next one 1 -- 1 1/2 ^2 * .4.

00:14:33.350 -- I have the right table. I'm just making sure I have the

00:14:35.318 -- right values OK.

00:14:37.450 -- And the next one 2 -- 1 1/2 ^2 * .2.

00:14:43.550 -- I have to move it down page or move it down the line.

00:14:47.400 -- 3 minus the mean squared times .1 and the last one 4 -- 1 1/2

00:14:53.715 -- squared times point. That's a two up there. Sorry times .1.

00:15:01.080 -- Well, you know this stuff and all these lovely little.

00:15:06.550 -- Squared differences in products all add up to 1.45.

00:15:14.620 -- Ann, if at anytime you're working through this on your own

00:15:17.887 -- and you get a different number than I do, don't hesitate to say

00:15:21.748 -- something. It happens, unfortunately, but it happens

00:15:23.827 -- and I won't be offended.

00:15:27.730 -- I used to wonder why I was like

00:15:29.498 -- why. How is it so easy to make mistakes? And I think it's

00:15:33.506 -- actually really easy on this end 'cause you get caught up in what

00:15:36.665 -- you're doing. You don't think about something that you're

00:15:38.852 -- dealing with right this moment when you're trying to talk about

00:15:41.525 -- something 5 minutes ahead of you know and think 5 minutes ahead.

00:15:44.441 -- Yeah, it's interesting, alright, but if I do make a mistake,

00:15:47.114 -- don't hesitate to let me know.

00:15:49.900 -- So. Of course, the variance leads us to the next one, which

00:15:55.212 -- is the standard deviation anisur standard measurement of spread.

00:16:02.230 -- And.

00:16:04.320 -- So it's a again a measure of spread.

00:16:09.480 -- Variation that's an R in there, sorry.

00:16:12.890 -- It is the average distance.

00:16:17.670 -- Without the squared.

00:16:20.910 -- Each point is from its mean.

00:16:23.690 -- Oh, there's an end in there.

00:16:32.600 -- So.

00:16:36.450 -- It's just the square root of the variance, and since it's

00:16:39.365 -- really what we end up wanting to do because its units of

00:16:42.545 -- measurement are the same as the mean, so they have single non

00:16:45.725 -- squared units of measurement.

00:16:47.830 -- It has seem.

00:16:55.120 -- Units of measurement azzameen

00:17:00.870 -- which is good? Want to keep things on the same scale?

00:17:06.180 -- And it literally is just the square root of the variance.

00:17:14.510 -- The positive square root, of course.

00:17:19.220 -- Remember, standard deviations invariances cannot be negative.

00:17:21.789 -- They can be 0.

00:17:23.810 -- Which is not very exciting, but they can't be negative.

00:17:27.280 -- 'cause if there is zero, you

00:17:28.678 -- have identical data points. Which I suppose is not

00:17:31.805 -- necessarily that it's not super exciting. There could be

00:17:34.550 -- a good case for it, but it might not be that exciting to

00:17:38.515 -- look at. So Sigma without the squared is our notation.

00:17:43.890 -- So SD of Y.

00:17:46.650 -- Probably, but as of why, but that might mean something else

00:17:50.005 -- in a different class, so I use SD and so we just take the

00:17:54.275 -- square root of our variance.

00:17:56.910 -- Or the square root of Sigma squared. Either way for us in

00:18:01.962 -- this example, is sqrt 1.45.

00:18:05.720 -- Which is fire Mario 1.2?

00:18:13.770 -- Or something close.

00:18:22.100 -- Alright, this would be great if we could always get population

00:18:25.070 -- values and we would never have to worry about doing. You know

00:18:28.310 -- we'd always be able to know everything about the population.

00:18:32.020 -- Not always the exact case in life. Unfortunately, there's a

00:18:36.010 -- lot of unknown.

00:18:38.650 -- And it's a two point 1.20 that some decimals off the end, but I

00:18:42.458 -- just found it to 1 decimal place as far as your work goes most of

00:18:46.538 -- the time using significant digits is not a horrible idea,

00:18:49.258 -- but I would say except in a rare case when we get to the last

00:18:53.338 -- chapters, you probably don't need to carry it more than two

00:18:56.330 -- to four decimal places are last chapters or there are some

00:18:59.322 -- concepts where we're going to have a small value, some sort of

00:19:02.586 -- density value which is very similar to like a growth or

00:19:05.578 -- decay rate, so you'd probably more like a decay rate, so you

00:19:08.842 -- probably want to make sure you might want to carry those out a

00:19:12.378 -- little bit further, but.

00:19:13.640 -- The most part significant digits or two to four decimal places

00:19:17.457 -- will be more than sufficient for what you need.

00:19:22.630 -- But unfortunately we don't have.

00:19:26.030 -- Population values all the time. He did. Life would be simple and

00:19:29.510 -- then we wouldn't probably need a whole discipline called

00:19:32.120 -- statistics for all this stuff because we wouldn't know the

00:19:35.020 -- entire population. But since we don't, we have to use

00:19:37.920 -- statistics. So what we're going to do is take samples and that's

00:19:41.400 -- really what you're doing. Here is looking at the samples from

00:19:44.590 -- surveys and what have you and trying to make estimations. Our

00:19:47.780 -- main estimations are going to be

00:19:49.520 -- a mean. A total which you may or may not have dealt with in your

00:19:55.123 -- intro class and a proportion. There are others, of course, but

00:19:58.764 -- those are our main.

00:20:00.180 -- Three statistics of interest while we're here in this course

00:20:02.840 -- and those would be the main three statistics of interest in

00:20:05.766 -- surveys as well. So.

00:20:08.360 -- And of course with that we always want to have a variance,

00:20:12.284 -- so we getting back to calculating this all right now.

00:20:16.590 -- You've probably seen that there are different.

00:20:20.540 -- Calculations formulas for population versus sample values.

00:20:25.880 -- So let's kind of take a peek at

00:20:28.336 -- the differences. Population.

00:20:33.650 -- Versus sample.

00:20:36.980 -- So population value for a mean.

00:20:40.470 -- Is mew. I'm going to write the word meaning here so we know

00:20:44.698 -- what this is at first case. It's been a little while since you've

00:20:47.402 -- seen some of these. So if we knew every single value in the

00:20:51.642 -- population, we would sum all of those up. So I'm going to use.

00:20:55.516 -- Not that you can tell the difference, but that's supposed

00:20:58.496 -- to be a capital Y versus a small way. Usually my capital wise are

00:21:02.668 -- straight just lines and my lower case. Why is usually kind of got

00:21:06.542 -- a curve to it?

00:21:08.830 -- I'll usually remind you as we get there, so we take every

00:21:12.514 -- value of the population.

00:21:14.700 -- And we divide it by. Now we have a new symbol, big End. Big N

00:21:19.875 -- represents population size.

00:21:22.480 -- I'm actually going to write that on my previous sheet of

00:21:25.351 -- paper that I'm going to bring down Tuesday here, so an is

00:21:28.483 -- always your sample size.

00:21:32.560 -- And Big N is going to be your population size.

00:21:37.320 -- Which is actually important. In this course we need to

00:21:39.380 -- know that for the surveys and stuff that we were analyzing.

00:21:45.350 -- Alright.

00:21:48.520 -- Before a sample.

00:21:51.400 -- Ala carte, why bar could be X bar. Yeah, Brooke. Uses why

00:21:54.856 -- we're going to stick with guys. We would take the sum of all of

00:21:58.888 -- our sample. Observations and divided by the number of

00:22:03.687 -- observations in our sample.

00:22:06.980 -- Depending on how we draw a sample, these could be

00:22:10.090 -- identical. But it's not going to happen terribly often, except in

00:22:14.340 -- my example. Today it was coincidence. I swear I actually

00:22:17.390 -- do a random sample, and the thing we're going to look at

00:22:21.050 -- today, and it turned out that the sample mean is going to end

00:22:25.015 -- up being exactly the population mean, but that doesn't always

00:22:28.065 -- happen, but it should be most of the time, pretty close.

00:22:33.250 -- Alright, variance.

00:22:36.060 -- So we call it Sigma squared.

00:22:40.420 -- I'm going to actually give you two different derivations

00:22:43.183 -- of the same formula.

00:22:46.410 -- There's one you've seen before.

00:22:47.860 -- Maybe sort of. So why that should be an eye for each

00:22:52.895 -- individual observation minus mu?

00:22:55.150 -- Quantity squared divided by big N.

00:22:58.680 -- Or in the discrete case, what you saw earlier?

00:23:04.730 -- That was the sum Y minus mu quantity squared times the

00:23:09.427 -- probability of Y.

00:23:15.730 -- Alright.

00:23:19.650 -- S squared following the same sort of formula over here.

00:23:24.040 -- It's going to be.

00:23:27.980 -- The sum why I -- Y bar quantity squared. We divide that by

00:23:33.661 -- little N -- 1.

00:23:36.370 -- Because it came from a sample and we're losing

00:23:38.773 -- some information. If you look there at the formula

00:23:41.176 -- I have. Why bar versus mu, since we don't know mu, we

00:23:44.380 -- lose. We lose our information. A degree of

00:23:46.516 -- freedom. So that's why we're dividing by N -- 1

00:23:49.186 -- little N -- 1.

00:23:51.980 -- But in the population case, we wouldn't actually lose any

00:23:55.010 -- information because we have it all. So, and we're using

00:23:58.040 -- the real mean.

00:24:01.050 -- This one here. Technically we still use via why, but it's more

00:24:05.154 -- ha with a hat on it, so anytime you see a hat on something

00:24:09.942 -- that's usually called an estimator and you're actually

00:24:12.678 -- going to see a hat on a V. More often than not. So this one

00:24:17.808 -- implies that we actually do

00:24:19.518 -- know. All the values and population. Here we are

00:24:23.030 -- estimating the variance, so it's the estimated variance of Y.

00:24:27.750 -- And it really isn't going to look hugely different.

00:24:36.970 -- As well, calculate the expected

00:24:38.450 -- value of Y. Or mew hat.

00:24:42.130 -- Yeah, this book likes to use hats on things, so if you

00:24:44.962 -- haven't seen that too much before, we're going to have

00:24:47.322 -- hats. Lots of hats.

00:24:51.340 -- Standard deviation, well, that's actually.

00:24:57.180 -- Not that exciting or different than what we were used to. So

00:25:01.212 -- Sigma is the square root of Sigma squared and over here S is

00:25:05.580 -- sqrt X ^2.

00:25:11.650 -- Alright.

00:25:15.800 -- Trying to keep my pages and pages in line here so in our

00:25:21.065 -- statistical studies we love to take samples and we make

00:25:25.115 -- inferences from those samples about the larger population. So

00:25:28.760 -- we want to make.

00:25:30.980 -- Well, it's an inference is an educated guess, but we're using

00:25:34.126 -- data and facts to back that up. So it is an educated guess.

00:25:37.844 -- Guess sounds so. I don't know Willy nilly versus.

00:25:42.350 -- An educated statement I don't know, but that's what we're

00:25:45.690 -- going to do. So a lot of times we want to make inferences about

00:25:50.366 -- unknown population parameters. So what do we do? We use our

00:25:54.040 -- sample statistics, so we're going to get back to our TV

00:25:57.714 -- example. 'cause it's completely exciting an in our TV example.

00:26:04.140 -- TV simple. Let's say I took a sample an it's not a very

00:26:09.301 -- big sample, it's only a sample size 4.

00:26:13.970 -- An out of this sample, we knew that we could have values that

00:26:18.169 -- were 0123 or four, but in this particular sample my values

00:26:21.722 -- were. Those are my sorry. These are supposed to be my curly

00:26:25.598 -- braces, but I suck at drawing them, so that's what it is.

00:26:30.900 -- These were my data points.

00:26:34.650 -- There we go.

00:26:37.580 -- 2013

00:26:42.340 -- now just for reference, our population had a sample or

00:26:44.780 -- had a size 4 as well, but we're going to take a sample

00:26:47.952 -- of size 4 and it could have been any values Now notice.

00:26:52.870 -- We actually had five different values that could happen. We

00:26:55.460 -- only chose for actually so big N is. I have a big I have a typo

00:26:59.604 -- on my thing. I gotta fix it big and is actually 5 here, alright?

00:27:04.520 -- So let's estimate mu. So mu hat. We usually just

00:27:07.920 -- call that Y bar X bar.

00:27:11.360 -- Pick a letter well, minus a few of 'em till pigsie.

00:27:17.060 -- But here it is. When we use the sum of our values

00:27:21.296 -- divided by your sample size. So we can do that.

00:27:27.960 -- Divided by 4, why are we doing it this way? Well in this case.

00:27:33.280 -- We're kind of assuming that they didn't have different

00:27:36.286 -- probabilities from our sample when we actually went to those

00:27:39.626 -- probabilities were different based on numbers in a

00:27:42.298 -- household, but from our sample, each of these had an

00:27:45.638 -- equal chance of being chosen.

00:27:48.740 -- So we do this and like I said before.

00:27:54.130 -- We actually get.

00:27:56.190 -- The same number, or pretty close to it, 6 force. I don't know.

00:27:59.986 -- Today is one of those days.

00:28:02.510 -- One of my favorite teachers in the math Department, so some

00:28:05.370 -- days are Calculator days, even for the most simple things like

00:28:08.230 -- 1 1/2. Which I already told you it was the same, but all of a

00:28:12.991 -- sudden my brain said no, you must test it again. So even

00:28:16.195 -- though I calculated it 2 hours ago, evidently I needed to do it

00:28:19.666 -- again. Alright now your sample mean is not always going to be

00:28:22.870 -- equal to your population mean. It should be relatively close

00:28:25.540 -- most of the time this just happened have been one of those

00:28:28.744 -- samples that I happened to draw and I did actually honestly draw

00:28:31.948 -- it randomly. And it just happened to be that this sample

00:28:35.507 -- mean was the same as a population mean, which is OK,

00:28:38.290 -- that's not a bad thing.

00:28:41.470 -- But now we're going to get into calculating our variance

00:28:46.070 -- and standard deviation.

00:28:49.020 -- So here's our variance. We can call it Sigma squared

00:28:52.170 -- hat or Sigma hat squared, probably Sigma hat squared.

00:28:56.380 -- Or you just call ask word that works too.

00:28:59.750 -- We're going to use the other formula, the second, well, the

00:29:02.335 -- first one I drew out, but not the first one we actually used.

00:29:06.540 -- Why I -- Y bar quantity squared divided by N -- 1?

00:29:12.260 -- That's the one we're going to use.

00:29:16.480 -- Amazon to all this lovely fun stuff.

00:29:22.350 -- And we got a zero. Remember using the values from the sample

00:29:25.878 -- and not the actual population.

00:29:28.710 -- 1 -- 1 1/2 squared and 3 -- 1 1/2 ^2.

00:29:38.620 -- Bye bye oh I was gonna say 4 -- 3. Now the answer is

00:29:42.904 -- three 4 -- 1.

00:29:47.900 -- We had five thirds or 1.67.

00:29:53.010 -- Versus what was it before 1.45?

00:29:58.000 -- So a little more variation in

00:29:59.842 -- this. Particular sample, then there wasn't a

00:30:01.996 -- population, that's OK.

00:30:05.390 -- And then for the standard deviation.

00:30:08.780 -- Sigma hat or S just take the square root of your S ^2.

00:30:15.100 -- Anne will get.

00:30:17.490 -- Our standard deviation 1.29.

00:30:22.170 -- As probably.

00:30:28.010 -- Alright.

00:30:30.940 -- Not very exciting, but I thought we do a nice little nice

00:30:34.816 -- overview. Just remind you so for random samples from infinite

00:30:38.046 -- populations, which is what we're kind of doing. The expected

00:30:41.276 -- value of the sample mean.

00:30:43.670 -- Is usually the true meaning that leads us toward what we're

00:30:47.850 -- looking at next, which is not just probability distributions,

00:30:51.270 -- but distributions of statistics.

00:31:02.590 -- Sample statistics so distributions of sample

00:31:05.278 -- statistics, or in other words, sampling

00:31:07.966 -- distributions. That's usually the more common

00:31:10.654 -- terminology.

00:31:16.340 -- So just a reminder, what a sampling distribution is that

00:31:21.235 -- it looks it's the distribution.

00:31:28.200 -- Of all possible samples, Whoops, there's 2 S is there?

00:31:37.310 -- Of a sample statistic.

00:31:46.050 -- We like that we have a specific theorem that we really really

00:31:50.106 -- like. And I need to go find that real quick here now. We probably

00:31:56.269 -- going to look through one of these on the computer up here,

00:32:00.985 -- but it didn't want to go

00:32:03.343 -- through. Well, I wanted to show I didn't want to necessarily go

00:32:06.891 -- through both of 'em 'cause the other one really just kind of

00:32:09.663 -- summarizes this whole thing together. So that's something

00:32:11.511 -- you can look at the other link for. It's called CLT 2.

00:32:15.230 -- But we're going to do is we're going to look at the sampling

00:32:18.961 -- distribution an. I actually have a couple of examples to

00:32:21.831 -- show through simulation how this actually works and why

00:32:24.414 -- we're still able to actually use a normal model. Most of

00:32:27.571 -- the time for analysis, and we're going to do a normal

00:32:30.728 -- model in this classroom as well for this course.

00:32:34.600 -- Not all your surveys are going to have variables that follow

00:32:38.131 -- normal models. OK, not all of 'em, but provided we look at we

00:32:42.304 -- have large enough samples and what have you most of the time

00:32:46.156 -- we should be OK, but not every time. There are exceptions to

00:32:50.008 -- that rule always. So first thing you should always do graph your

00:32:53.860 -- data if you don't know what your data looks like visually, then

00:32:57.712 -- you're only getting probably about 1/3 to half of the

00:33:00.922 -- picture. So alright, so we're gonna look at the central Limit

00:33:04.453 -- theorem. And for that one, our sampling distribution of the

00:33:08.129 -- sample mean is approximately normal with a mean mu and

00:33:11.539 -- standard deviation of the sampling distribution of the

00:33:14.267 -- sample mean. Is Sigma divided by square root of N. So since

00:33:18.359 -- we're looking at the distribution of the sample

00:33:21.087 -- mean, we don't just use our variance, we take the variance

00:33:24.838 -- divided by N or the standard deviation divided by the

00:33:28.248 -- square root of N. We call that Sigma over square root of N.

00:33:32.681 -- We used to call that a standard error.

00:33:36.760 -- That is provided that N is sufficiently large. This theorem

00:33:39.570 -- can also apply to other statistics, which is really,

00:33:42.099 -- really handy because we're going to be using those other

00:33:44.909 -- statistics as well. The sample proportion an one of 'em I

00:33:48.000 -- didn't actually have on here. The sample total which could be

00:33:51.091 -- used in case I don't know if you guys have ever dealt with the

00:33:55.025 -- total before, but it could be nice, say for an airline we need

00:33:58.678 -- to know how many passengers are boarding the plane right? And

00:34:01.769 -- the other thing we do is we weigh how much your bags weigh.

00:34:05.810 -- We need to know the weight of your bags, how much junk

00:34:09.002 -- you're taking with you on the plane, in addition to the

00:34:11.928 -- weight of everything else on the plane, the humans on the

00:34:14.854 -- plane, everything.

00:34:16.640 -- So it might be nice to know what the average weight per person

00:34:20.085 -- should be. The maximum average weight per person, but that's

00:34:22.735 -- not the only thing of interest. It could actually be of interest

00:34:25.915 -- to look at the entire plane full of people's total weight. That's

00:34:29.095 -- just one example. It's not the only one, but it's one of the

00:34:32.540 -- few examples that you could use a total for, and so that's how

00:34:35.985 -- that's going to play in when we start getting to that.

00:34:39.770 -- Alright, so for the most part, the sample size should be

00:34:43.752 -- approximately at least 30.

00:34:46.100 -- If your distribution wasn't already normal to quote

00:34:48.956 -- unquote, guarantee the normality I say and kind of

00:34:52.169 -- using that term guarantee a little. Loosely, there's no

00:34:55.382 -- guarantees, but to get us the approximate normality,

00:34:58.238 -- we should have a sample size of at least 30. Now, if your

00:35:02.879 -- original distribution you already know is inherently

00:35:05.378 -- normal, that sample size stipulation is not required.

00:35:08.234 -- You could have a sample size is smallest 2.

00:35:12.850 -- But if you don't know anything about your original

00:35:15.613 -- distribution, always safer to take a sample size of at least

00:35:18.990 -- 30. That being said, in surveys we take, sample size is usually

00:35:22.674 -- of probably at least 10 or more times than that than 30, so.

00:35:27.480 -- And we're going to sample proportion. We usually want to

00:35:30.000 -- sample size of at least 60. Most of your information from sample

00:35:33.024 -- surveys, alot of time, not most or all. But a lot of times are

00:35:36.552 -- going to be percent, so that would be of interest.

00:35:39.800 -- And again, I said here, if you're just distribution is

00:35:42.520 -- already inherently normal, your sample size stipulation can be

00:35:44.968 -- ignored. It's not that you're ignoring it, but it's not. It's

00:35:47.960 -- not relevant to what you need to worry about it, alright?

00:35:52.040 -- This one sorry. The book I was using used pie instead of P for

00:35:56.184 -- the proportion. Now it should be like most. I'm an intro books,

00:35:59.736 -- they always use P, but as soon as you hit like our 431 class,

00:36:03.880 -- that book uses pie 'cause everything else uses a Greek

00:36:06.840 -- letter. Why not? So why not intro class? Well unfortunately

00:36:09.800 -- will never find that answer out but we still go back to P in

00:36:13.944 -- this book. This book uses P for that terminology just to kind of

00:36:17.792 -- let you know. But you can interchange it with pie. It is

00:36:21.344 -- the same basic thing.

00:36:23.990 -- Alright, so in shorthand notation. Our sample mean X bar

00:36:28.110 -- or why bar is distributed normally with a mean mu and the

00:36:33.054 -- Sigma sub X bar is another notation for that standard

00:36:37.174 -- error. Sigma over square then.

00:36:41.150 -- Same thing for the one for the proportion and the total would

00:36:44.042 -- work as well. I thought this was my updated file that showed.

00:36:47.630 -- Totals so all this is nice and interesting in review. You'd not

00:36:51.950 -- be calculating Z scores in here.

00:36:54.890 -- So if you're hoping to see Z&T scores in here, I'm actually

00:36:58.454 -- going to see those, but that's OK, Alright? This is the

00:37:01.721 -- important part, so we actually see how this distribution works

00:37:04.691 -- and how the central Limit Theorem helps us to look at

00:37:07.958 -- normality. So we're actually going to look at a distribution

00:37:10.928 -- that's already normal, so it's not going to be that exciting

00:37:14.195 -- when we take the look at the sampling distribution, it's

00:37:17.165 -- still going to be normal. There are going to be some

00:37:20.432 -- differences, but then we're going to look at an exponential

00:37:23.402 -- distribution, which is obviously

00:37:24.590 -- not. A normal Bell curve distribution and a binomial

00:37:27.672 -- distribution, just so you can see how the central Limit

00:37:31.062 -- theorem works on even the non normal distributions.

00:37:35.200 -- You don't ever have to reproduce this unless you want to, and

00:37:38.608 -- which case if you want to borrow my code, just ask me, But what

00:37:42.584 -- this does is I'm basically going to take this is in our command,

00:37:46.276 -- so our norm. And you plug in how many values you want into that.

00:37:50.969 -- That will give you random numbers generated from a normal

00:37:53.659 -- distribution. If you don't specify the mean and standard

00:37:56.080 -- deviation, it will assume the mean is 0 and the standard

00:37:59.039 -- deviation is 1, just like the Z

00:38:00.922 -- distribution. So we needed.

00:38:04.070 -- And in this case I actually gave it a different mean in a

00:38:08.256 -- different standard deviation than the Z distribution. So I

00:38:11.154 -- took a sample of 500.

00:38:13.810 -- Out of a normal distribution and I set the mean at 100 and the

00:38:18.612 -- standard deviation at 10 and I said, oh, let's look at the mean

00:38:23.071 -- so mean for that particular sample was 100.25.

00:38:26.970 -- So close.

00:38:29.170 -- And here's our histogram. So the spread on this one goes from

00:38:33.898 -- about 65 to 135, give or take.

00:38:39.630 -- And another random sample just to show the mean change

00:38:42.480 -- to her. But we're still right around that 100 mark.

00:38:46.440 -- And then. For some silly reason, I decided I need to put a curve

00:38:51.432 -- on it. I hardly ever put curves on my on my distributions like

00:38:54.890 -- this, but this one was like I'm going to put that curve on

00:38:58.348 -- there. So there it is. So it is a normal distribution still

00:39:01.540 -- spread out between 65 and 135 center right about 100.

00:39:05.510 -- Oh, rest of my code fell off, sorry.

00:39:09.260 -- Alright, so for this simulation process I'm setting the mean and

00:39:12.714 -- the standard deviation. I'm going to take samples of size

00:39:16.232 -- 5 and I'm going to do that 500 times. We're going to have 500

00:39:20.236 -- samples of size 5, so we can look at the means of all of

00:39:24.240 -- those, and that's what I'm calculating here.

00:39:28.350 -- And then we look at histogram and there is the distribution of

00:39:32.898 -- the sampling distribution of the sample mean. So the spread

00:39:36.688 -- changes 'cause we're dividing it by the square root of N. So it's

00:39:41.615 -- now spread from about 85 to maybe 115 versus 65135.

00:39:46.160 -- So the curve got skinnier and a little bit taller and that

00:39:50.324 -- happens. But it's still a normal distribution, but

00:39:53.151 -- this is now the distribution of X bar versus X.

00:39:56.910 -- And they are just kind of arbitrary values. I guess I

00:39:59.968 -- just. Grabbed grabbed a mean in a standard deviation and

00:40:03.884 -- just used it so.

00:40:05.900 -- Normal spread change though.

00:40:08.720 -- That's important to look at.

00:40:10.480 -- Exponential distribution. I don't know why I really like

00:40:12.919 -- this distribution. If you took 201 or 251 then chances are you

00:40:16.171 -- probably didn't see this. You may have heard about it, but you

00:40:19.423 -- probably didn't see this. If you take 301, they may have seen

00:40:22.675 -- this, but don't stress it if you

00:40:24.572 -- haven't seen it. I'm not going to test you on this formula,

00:40:29.394 -- but this just shows you the formula I'm using, so it's an

00:40:33.786 -- exponential distribution. Exponential is really great

00:40:35.982 -- for modeling the waiting time between events.

00:40:39.620 -- Other processes too, but that's one of its big draws.

00:40:43.920 -- Now let's see. Here we are going to be looking at this with this

00:40:47.784 -- one. We're going to use a distribution with a rate of 1.

00:40:52.180 -- Alright, so random number again, a different

00:40:54.595 -- distribution R has told whole bunch of different

00:40:57.355 -- distributions. You can randomly generate numbers

00:40:59.425 -- out of which is great.

00:41:03.170 -- We need N.

00:41:05.160 -- Anna rate. So with this one we're going to sample size 500.

00:41:10.040 -- We will find the mean.

00:41:11.880 -- That's pretty close to 1.

00:41:14.470 -- That sample #1 sample #2.

00:41:18.450 -- Actually knows same sample. This sample number one.

00:41:21.362 -- Sorry, obviously not a normal distribution.

00:41:25.370 -- And do it again. This time the mean was to even just a hair

00:41:28.968 -- lower. But we're still pretty close to the one mark.

00:41:33.720 -- There we go. Being silly had to add that curve in again. So

00:41:37.984 -- there's our curve or exponential curve and the

00:41:40.936 -- regular distribution of it.

00:41:43.260 -- So now we're going to do the same thing, except for I'm

00:41:46.776 -- going to be taking samples of size 30 and I'm going to

00:41:50.292 -- take 500 samples of size 30 to calculate. 500 means joy.

00:41:54.530 -- It's kind of fun to do. Well, this is the first time so.

00:41:58.840 -- This one, a sample size of 30 almost gives it the normality.

00:42:03.016 -- It's not perfect, but it's.

00:42:05.570 -- Close enough, that's the one thing that's hard to once you

00:42:08.309 -- get out. Intro class is looking at some of these

00:42:10.799 -- curves, and some of these they might not look normal to. You

00:42:13.787 -- might want to go. Some of these are going to be normal

00:42:16.775 -- enough. This one is actually good.

00:42:19.450 -- Obviously not exponential anymore and then binomial. So

00:42:23.266 -- remember binomial distribution is one of those discrete

00:42:27.082 -- distributions for absence or presence, so success or failure.

00:42:33.220 -- So this one is again 500 samples with a binomial

00:42:38.020 -- distribution. Its probability of success was .8 an. We did

00:42:42.820 -- sample sub size 10.

00:42:46.370 -- But this person will probably do 500, though again binomial. You

00:42:49.967 -- can randomly generate, so this first one is actually 500. Later

00:42:53.564 -- on when we do, the 500 samples were going to take 500 samples

00:42:57.815 -- of size 30. I think or is it 10, probably 10? I don't know. I'll

00:43:02.530 -- double check. I looked through it today and then I forgot.

00:43:06.200 -- So the eight the mean should be 8, so the mean for a

00:43:11.127 -- binomial is N * P, so 10 times .8 gives us 8 and this one's

00:43:16.812 -- pretty darn close 7.98.

00:43:20.370 -- Not even a continuous distribution.

00:43:25.820 -- There we go again, and this one that means just a hair over

00:43:29.668 -- eight. OK, so that was our second random sample and there's

00:43:32.924 -- our second. Histogram.

00:43:36.780 -- Same process we're doing samples of size 10, but

00:43:39.012 -- we're taking 500 of them.

00:43:42.220 -- And look at that all of a sudden. It's not the prettiest

00:43:46.084 -- thing I've ever seen, seen prettier distributions, but

00:43:48.660 -- it's still approximately normal.

00:43:51.700 -- Excuse me, centered right about 8:00, so that's what the central

00:43:54.989 -- limit Theorem does. Remember, when I took my intro course, I

00:43:58.278 -- was just like it was just kind of this concept. You had to just

00:44:02.464 -- think about it was like, OK, I'm sure I'll use it, but actually

00:44:06.351 -- saying it for me it made a huge

00:44:08.743 -- difference this other. Thing that you get Lord death right?

00:44:12.372 -- I zoomed in, sorry this other one that you can look at is

00:44:16.090 -- just moves a nice little handout that my 200 level class

00:44:19.236 -- professor had given to us. So I asked him if I could steal it.

00:44:23.240 -- Well, I said I asked him if I could borrow it so I said well

00:44:27.530 -- can I. Can I borrow it and give it to my class and you said OK,

00:44:32.106 -- that's fine so I stole it. There it is but I did I did put

00:44:36.396 -- his name down there so.

00:44:39.460 -- Alright. So we're looking at this thing. We're probably not

00:44:43.222 -- gonna be able to finish this up today, which is OK. We can

00:44:46.576 -- finish this up later, but we can kind of set ourselves up for the

00:44:50.188 -- end of this. So what we want to do?

00:44:54.510 -- Is we have our population at see here.

00:44:59.810 -- I left my pen open, sorry.

00:45:03.800 -- So this is our original population values.

00:45:08.530 -- And we're going to.

00:45:11.320 -- I think in this case just take samples.

00:45:19.530 -- Size 2

00:45:22.310 -- just keep simple.

00:45:25.620 -- Now.

00:45:28.390 -- In this case.

00:45:31.100 -- This is this is our population and this is the number the

00:45:35.132 -- sample size we're going to do. We want to actually look at all

00:45:39.500 -- possible samples for this so.

00:45:49.860 -- All possible samples.

00:45:54.600 -- From in this case, what we're doing is those were the number

00:45:58.344 -- of TV's in the House, but what we're going to do is we're going

00:46:02.712 -- to be looking at from 4 houses.

00:46:06.240 -- So let's say we have House 1-2, three and four.

00:46:13.590 -- So this has a population of four different.

00:46:19.210 -- Possibility so for houses small town. There we go

00:46:22.666 -- more than Moscow.

00:46:26.250 -- One of the things we got excited about when I was a kid

00:46:29.396 -- we were driving. I think we were driving to California and

00:46:32.058 -- we were driving through southern Idaho really late

00:46:33.994 -- tonight. My dad got all excited how to wake all of us up. It

00:46:37.382 -- was like 3:00 o'clock in the morning. 'cause one of the

00:46:40.044 -- towns we came from California so this was a pretty cool

00:46:42.706 -- concept to us. 'cause it was cute, neat. One of the towns

00:46:45.610 -- actually like listed on the animals and I can't remember

00:46:48.030 -- what town it is but listed all the animals, the cows, the

00:46:50.934 -- humans telling my dad had to wake us all up. Look, look at

00:46:54.080 -- this look at this.

00:46:57.060 -- Alright, so small town that was a small town, not as small as

00:47:01.038 -- this little town we're going to deal with, so we're going to

00:47:04.710 -- sample the houses. And then we're going to ask them.

00:47:11.280 -- How many?

00:47:13.980 -- TV's do you own?

00:47:20.060 -- All right, we're going to look at all possible samples.

00:47:25.100 -- So if we just line 'em up.

00:47:29.080 -- One and two can be one of the samples 'cause we're

00:47:31.621 -- taking samples of size 2.

00:47:34.540 -- Now we're obviously going to assume something here that's

00:47:37.528 -- going to be kind of important for us to talk about. Kind of

00:47:41.844 -- important. That's an understatement.

00:47:44.660 -- Is that we're doing this?

00:47:49.350 -- Without replacement.

00:47:52.210 -- So what I'm doing here is that when I choose a house,

00:47:56.086 -- it can no longer be chosen for the observation #2. So

00:47:59.639 -- if it's been chosen for observation number one, it

00:48:02.546 -- can't be chosen again for observation #2, so we

00:48:05.453 -- couldn't go to the House number one twice or House

00:48:08.683 -- number 2 twice, etc.

00:48:12.680 -- Now two and three can be chosen, and this is all

00:48:15.969 -- possible samples. It's not what we actually did, but

00:48:18.660 -- we're looking at the possibilities.

00:48:21.830 -- I don't know. I like Roman numerals. I always have it

00:48:24.888 -- thing from when I was a kid. I apologize, but I'm not

00:48:28.224 -- that sorry.

00:48:31.500 -- So these are all possible samples. We had six of them.

00:48:37.480 -- And that's where we get to pick

00:48:39.545 -- up next time. Figure out what to do with this thing.

00:48:46.780 -- And that's it, and we will finish this tomorrow or

00:48:49.910 -- finish this next class.