We're talking about how do you actually make money on YouTube here on the journey? All right, so we're answering your questions about how to make money on YouTube and for that I have a special guest William with GoDaddy. Hey guys. Yeah, so I do a ton of stuff with marketing YouTube included Advertising have helped businesses grow from 20,000 4 million and 13 months crazy stuff, right? I love marketing I love figuring out how people can tell their stories writing about this on entrepreneur and obviously on the GoDaddy blog And I'm excited to dig into some of the things that I was able to uncover With some of the big you YouTube influencers that I was able to reach out to that's awesome So I think we first have to ask ourselves. Why do people want to go to YouTube to make money? What is so attractive about it? Give me a Yeah, it's all yeah. Sorry. That's it. No 26 million though. That's how much Ryan's world made last year alone Okay, that's grossing channel on YouTube That's the little kid that started off just doing toy reviews, right toy reviews, right? When you think about that though when we were kids we were thinking about hey, I want to be in the NBA I'll get that later We want to be in the NBA or something right because that's where we're gonna have our big aspirations of making millions and millions of dollars But for a new generation This is their path towards that wild dream of being a multi-millionaire, right? Like anybody can get started and make videos on things that they love, right? There's kids that out there a little dad and son combo that cuts things in half and makes millions of dollars just sing What's inside stuff? It's crazy. Yeah, there's a lot of examples of that and what's nice is it doesn't have to be just the People are that are the most physically gifted to be in the NBA or something like that This could be a niche of you know, whatever that is that you're into which is exciting is for something for everybody So the question is can you actually make money on YouTube? The simple answer is yes, right? Right? You can and while we kind of just said that it's very simple. It's also incredibly complex There's a lot of things that you need to know. It takes a lot of hard work. It takes dedication You need to keep doing it, you know consistently find who that audience is and that's the part that we're gonna Hopefully help you walk through on this video and simply showing up isn't enough You've got to be able to show up in the right way at the right time to the right crowd You did a little bit of pre-work before this video to help out our audience I wanted to make sure that it wasn't just based on, you know My thoughts or things like that because while I do a lot within the marketing world I'm not super deep into YouTube to the point where I am one of those people making millions of dollars on YouTube So I wanted to get information that was relevant from the people that are actually doing this on a daily basis Making a living from this In the first question that I wanted to ask them when I reached out was what is the one tip? The best piece of advice that you could give to somebody that wanted to make money on YouTube Do not go into it trying to make money go into it because you're passionate about something I read something recently about how influencers are not they're not just like I want to have a following on Instagram They do something cool and they want to share it with people and they want to teach people So if you're gonna start a YouTube channel do it because you're passionate about what you want to do Sorry, I'm talking so fast. I just get so excited and then Share that with people share the story share the journey teach people be productive in other words Don't expect to make just one video or 12 videos or even a hundred videos. You got a really Knuckle down and be productive, right? It's gonna take more than you think Be interesting obviously be entertaining, but also be interesting in other words. What would you be interested in seeing? The most successful and monetized YouTube channels I notice Generally have something very interesting and that they've researched or that they offer Whether it's a tour of the world or explaining something that's kind of difficult to understand and in science or whatever These are fascinating channels and they're they're interesting not because they're eye candy Although slime is very popular Last time I looked is they're useful in other words You can they're educative you can learn something from them and you know impress people with your knowledge also be Be creative, you know, there's a lot to be said for just coming up with an idea and the most Viral videos I have have been extremely creative that have been spawned just from my own Interest in different celebrities different voices different poems different viewpoints that I want to express and so that's good and Be patient Because even though we have a very fast-moving culture and YouTube is very fast and everything's going very very very fast we got to be patient because like I said, you got to keep putting things out and It's not gonna happen. It's not gonna happen overnight or if it does Then what are you gonna do next week after it hits great video there awesome advice And I love that he called out that you have to be Interesting but also useful. I think we try to just create content just create content It doesn't really provide value to the audience. Yeah, there were two things that I really liked about what he said And one was the idea of of you know knuckles down because it talks about that hard work but like you said the most important thing that I take away from that is First we found out what your why is for why you're gonna do this But even more important than your why now is why does this viewer care? Right, what kind of content are they looking for why do they care to care to hear from you? Some people are looking for an escape. So one of the channels that we looked at You know, she's documenting things that she's doing all over the world and you can almost feel like you're you know Going around the world with her another one that we're gonna look at later on They are helping people learn Spanish and so some people are looking to learn something But ultimately there's a reason why somebody is trying to watch that video and think about that person Because that's what's going to help your videos connect, right? And I think what a lot of people get stuck in the trap of the sales mindset when they create content They're doing it to try to sell something and if that's what you're going YouTube doing to make it rich and just try to sell Sell sell you're not creating any value and no one's gonna want to watch that Value first to get through the viewership in there and then you can lead in different things We'll talk about that a little bit later The other question that I wanted to dig into was the idea of I don't like Having to learn from my own mistakes if I can learn from your mistakes or somebody else's mistakes It's less pain for me, right? So the other question that I wanted to ask them is what is the biggest challenge that you faced when you're trying to monetize your YouTube channel? The biggest issue that I've had that's trying to monetize content on YouTube is the changing content guidelines YouTube is always changing its advertiser friendly guidelines So it's best to stay on top of that and know what kind of content you can and can't make and if a video does get Demonetized or deemed unsuitable for all advertisers not only do you lose out on ad revenue But also potential viewers because the algorithm is less likely to recommend a video that's been deemed unsuitable for advertisers See I didn't even realize that with creating content on YouTube that you have to kind of stick to those guidelines because they can change and then Deem your content unworthy then it's not shown to anyone, right? And it's not even so much that they can change It's that they will change again. It used to be just 10,000 views now It's four thousand watch hours and 1,000 subscribers So they've significantly made it harder for people to get to that point where they can begin monetizing on the YouTube platform We really want to talk about the algorithm Did you know that 70% of the videos watched on YouTube are due to the algorithm themselves? Well, it makes sense when you think about it from their perspective. So YouTube's a business as well They're trying to make money and if they change the algorithm They're changing the algorithm based on how people are watching things But now when they've changed the algorithm if they can't monetize the video that you have all of a sudden YouTube's algorithm is going to deep moat that video. It's not going to show up as high in the normal search results that people are doing So now that type of video content is just less relevant, right? I mean at the end of the day YouTube's main goal is to keep you on YouTube So if you watch a video and then have a video that's similar that you're like, oh cool I'll check that out and I'll check that out and I'll check that out Kind of put you deeper down the rabbit hole of that algorithm and then with that the content itself You really want to think about the algorithm when you create your content. What are the latest trends that are happening? I know there was a big update with mid-rolls So you had everyone just jumped on board started making 10 minute videos because they had to have that mid-roll It might change in the future and then you'll want to jump on that trend to really get on the algorithm first Make sure you're monetizing your videos, right? And we see this not just on YouTube but all sorts of channels, right there for a while You know, I don't know how much you guys keep up on some of the other trends But you know certain key words were really hot being in like a like a post title, right? And it was like clickbait, but then algorithms started catching on to this clickbait not driving actually good content And so the algorithms change to promote stuff that people actually want to read and experience and enjoy and watch on YouTube So touching base on your mid-rolls wait, you're touching what not well the mid-rolls in the actual YouTube ads That's another example of where all of a sudden certain people had access to those but not everybody did And so we kind of call that adpocalypse where certain let's say video creators have access to other features and ways to monetize That some smaller advertisers don't have so it's important that you keep up with all the different trends on what's going on with that Algorithm because that's what YouTube is looking for. Yeah Not only do you need to be up-to-date with the trends you also need to be up-to-date with the rules and policies too good point Yeah with the latest update as of right now when we're filming this video You can't market to kids anymore. So when you create your content you have to mark Is this geared towards kids or is it geared towards adults because it is there's gonna be no advertising for them And it ultimately it's to make the internet a bit more of a safer place But you have to make sure that you're aware of these things so it doesn't hurt you Another technique that I like to use when I'm trying to figure out how to do something is Deconstruction looking at something that was successful in figuring out. How did they do that? Why what made that? Successful so the other question that I wanted to ask these influencers was just what is your most profitable video? And why do you think it was so successful? Yeah, let's check it out So our most popular video on YouTube is how to make small talk in Spanish It has almost six hundred thousand views. Why do you think it is our most popular one? I think it's because we were able to dominate the SEO search results for some of the terms in the title of that video So search engine Optimization has a lot to do with your success on YouTube if you're making videos that are Answering questions that people are searching for often times They'll show up in Google or YouTube itself when people search for those terms This can help generate a lot of organic traffic for your videos that is huge So many people don't realize that YouTube is the second most popular search engine in the world and many of the creators Aren't SEO experts So if you take some time and invest in the SEO of the titles you use of the descriptions the keywords and things like that You can dominate that search engine Well, and I think that's the important part here is that there's two elements to SEO, right? There's the technical side making sure you have the right tags and keywords in your post But that's only one part of it and it's important part.  But once you've got that to really start deconstructing let's say the tyranosaurus rex You've got to think about like well Why does it have these teeth and what was it doing with those teeth and how you know? Why these short arms and stuff like that? That's the rest of this and I'd say that's probably the more important part of SEO And I look at that. It's just simply saying what are people asking start there when you you know You're trying to make a video about how to learn Spanish Well, what are the things that people would ask when they're ready to learn Spanish? It's probably not you know top 10 tricks for Spanish Maybe it is but most likely it's gonna be something along the lines of you know How do I say this or you know, what about this? Yeah, and that's crucial. So there's two couple resources that I have for you One is go on Google and start typing whatever the general keyword is Google will try to pre-fill the rest of the phrase and You essentially have your answer right there There's also answer the public comm where you throw in your keyword or your phrase and it'll give you Relevant keywords and questions that actual people are yet basically asking online You can tailor your but your content to that the good news here though Is that you don't have to rely just on? Monetizing through YouTube There's a lot of ways that you can make money from the videos that you do on YouTube when it comes to profitable content Almost every video on my channel makes some kind of money However, the things that generate the most money on my YouTube channel are my live streams And the reason for that is because we have people's attention from anywhere from three to six hours Almost every single week and during that time We have a bunch of different questions that we're answering about YouTube and over and over and over again a lot of the tools And things that youtubers use that I get a affiliate commission off of are linked to in the video descriptions of our live streams And because of that it makes it easy for people to find when we're talking about them And we present them as solutions to the problems that people commonly have on YouTube in addition to that our live streams Also generate super chats. They generate ad revenue and at a higher level our live streams Because we sit there and we prove for three to six hours straight that we understand our subject matter because of that It creates additional opportunities outside of YouTube as well that are also money-making opportunities So because of that the live streams hands down are the biggest money makers that I have on my YouTube channel Yeah, and what I love about that video is he was essentially diversifying his revenue stream It's not all coming from the advertising itself and a way you can do that is with YouTube premium views It's similar to patreon where you have your viewers pay a monthly subscription fee of whatever amount and they get extra Exclusive content only for them. Yeah, and there's even more things even within YouTube as far as super chats Live streams. There's a lot of opportunities on the YouTube platform to monetize But there's also a lot of opportunities off of the YouTube platform to monetize And this comes down to just being creative about how you go about that, right? So one example is you might be an e-commerce store Well, you can use this to showcase your products and then end up sending people to your website to convert Right and that's he talked about affiliate links He if you're creating a video then it makes sense to say hey check out the product that I'm using right here that I'm reviewing here's a link down below you get a little cut behind that or I know a lot of people will Basically create swag or shirts or stuff of their own brand to sell directly on YouTube or on their website, too Let's find something that's really exciting to your audience and that might be depending on your content I know Brian Dean does a lot of this with SEO Well, he has courses and you know ebooks and things like that that you can download that you can buy that you can use That for his services And and purchase those things as well absolutely and I was gonna bring up video influencers They're great brand talks a ton about how to create awesome videos They also have links to their courses and their ebooks and all their other content It's just adding more value because anyone can make a shirt But if you give extra content like that that actually changes someone's life that uses it That's where you'll win and again just to break it down diversify your content Add value to the people who are watching this and make sure that if you do sponsored content follow the FTC guidelines So I got a lot of great responses from so many people with amazing information But I think the one that I want to leave you with today is this one My best piece of advice for people looking to bank money on YouTube is to just start So many people get frozen by the creative process and think what they have to put out It has to be perfect. It absolutely doesn't have to be so jump in get involved have fun and find your voice first I love that just start we get so caught up with what are people gonna say? Am I gonna have the best video is gonna be perfect just do and get better as you go on practice makes progression True I think for just what everything in life, right? Sometimes you just have to get started All right, you are ready to make some money on YouTube Make sure you like this video add a comment below Subscribe to our channel and ring that bell see these episodes first. This is the journey. We'll see you next time


So I want to take some time to talk about Marianne Williamson's appearance on MSNBC. She did a phenomenal job. She basically said everything that I would have said if I had the opportunity to talk to MSNBC's audience. And I know what you're thinking. If you're watching this channel and you're a leftist, you already know that MSNBC is trash, right? They're basically the propaganda arm of the Democratic Party in the same way that Fox News is the propaganda wing of the Republican Party. Still, this channel is important because they are trusted with the Democratic Party's base. Normie Democrats love MSNBC and they take what MSNBC hosts say as gospel, essentially. So if we could get through to them and get them to at least start to think a little bit more critically about the Democratic Party establishment and so called moderate Democrats, then perhaps they start to warm up to the idea of supporting more progressive candidates and presidential primaries. And we don't just get our clocks cleaned because MSNBC says Bernie bad or the next progressive bad in upcoming presidential primaries and primary races across the country more specifically. So Marianne Williamson explains exactly why Biden is failing. And MSNBC Democrats, they know that there's all these excuses. Joe Manchin, Kirsten Sinema, the parliamentarian. So they might instinctively want to cleanse Biden of any culpability whatsoever. But what Marianne Williamson is forcing them to do is to think more critically. There's a number of things that Biden can do unilaterally via executive order just with his pen, but he's not doing it. And that's because he doesn't care and he's also corrupt. So Marianne Williamson makes a very strong case as to why Biden is a failure. And this is really important because, again, remember who she's talking to. She's talking to MSNBC's audience, which is just overall unapologetically loyal to the Democratic Party. This is really an important message that they desperately needed to hear. Let's watch. Marianne Williamson joins me now. Marianne, it's great to have you on the show. Thank you so much for your time. Just last month, you endorsed a progressive primary challenger to long time Democratic Colorado Congresswoman there. And when political raised the subject of you possibly challenging Joe Biden in the 2024 Democratic primary, you declined to comment. You clearly have issues with the Democratic establishment. Tell me a little bit about what role you are looking to play politically. Well, the role I most want to play politically right now is citizen. And as a citizen of this country, obviously, I'm very concerned about the 2020 elections, because, as you said, the president's agenda is not succeeding in high approval ratings and getting him high approval ratings. It is not doing the kinds of things for the American people, which could legitimately be believed to make people want to stand in line for seven or eight hours as they did before. The president came into office with promises, promises to help people, promises to lift them out of their misery. And that wasn't just a promise if Congress will allow me to do it. So I think that the promises to fulfill the will of the American people and to help the American people can still be accomplished even without congressional approval of such things as the Voting Rights Act or such things as Build Back Better if the president will use the power of executive orders. He could eradicate that college loan debt today. He could get 36 million people. If he would just take $50,000 out of each one of those loan debts, 36 million people would be unshackled from those debts. They could start businesses. They could own homes. He could use a national medical emergency to basically give Medicare to every single person in America. He could do that tonight. He could declassify marijuana from a Schedule I drug. Right now, I'm just a citizen who wants, as any Democrat does, and I think any reasonable person wants, to forestall the kind of red wave that could be coming at us in 2022. Because remember, it's not just about a Republican win. It's about what kinds of Republicans stand to win. And we have 435 Democrats who are running for the House. We have 34 Senate races going on in 2022. Those candidates need to be able to say to their constituents, the Democrats will help you. And right now, the Democrats are not helping people the way we need to. You never hear that on mainstream media. And that is so important because for the very first time, an MSNBC viewer who never actually thought about the Democratic Party and the way that they're failing might be thinking, wait a second, that's kind of a good point. I didn't know that Biden can simply cancel all student debt via executive order. And so why isn't he doing that? I have a child that went to college and is strapped with student debt that I didn't have to pay when I went to college. Why isn't he doing that? That's really interesting. Wait a second, he can actually give everyone healthcare in America, but he's not doing that. Why isn't he doing this? It's just really a matter of planting the seeds, which doesn't get done on MSNBC. To the extent that they allow left-wing political commentators on the network, such as Sam Cedar, I mean, it's pretty rare. So when we see Marianne Williamson go on MSNBC, David Cerota, Sam Cedar, I think that that's really important. These are valuable appearances that help to, for lack of a better word, radicalize MSNBC's normal audience. Now, everything that Marianne Williamson is saying is absolutely correct. We've talked about how he can cancel student debt on his own, but a lot of people don't even know that Joe Biden actually can give everyone in America Medicare for all because of the Affordable Care Act and a specific provision within that legislation. Now, it's a little bit complicated, and this legal maneuver would unquestionably be challenged by the courts, the pharmaceutical industry, the health industry wouldn't allow this to stand. But still, if Joe Biden cared, it's at least worthwhile to put up this fight, but he won't even do that. So if you don't know, let's actually talk about this. So this is in the American prospect written back in November of 2020 by David Dayan, and he explains the people of Libby, Montana, population 2628, share something in common with the rest of the developed world, but not their compatriots in the United States. They all have access to a single payer Medicare for all system. As part of the Affordable Care Act, residents of Libby who were exposed to hazardous airborne asbestos from a vermiculite mine owned by the W.R. Grace Company were made eligible for Medicare for free at the discretion of the Social Security Administration and the Department of Health and Human Services. It was codified in section 1881A of the Social Security Act. The language of the statute refers to any individual subject to an environmental exposure, though it was well understood at the time that this was about Libby. There's an environmental health hazard spreading through the entire country right now. It's infecting people unsuspectingly and killing hundreds of thousands. It's bound to saddle those who survive with long term and potentially debilitating health consequences. The incoming Biden administration can give all 11 million people infected with COVID. And if they really want to be aggressive, all Americans who have tested positive for coronavirus, the option of free Medicare coverage immediately. The primary pilot program in section 1881A is the Libby Montana program, but the executive branch is also given the authority to establish optional pilot programs to any area subject to a public health emergency declaration. The entire United States has been under a public health emergency since January 31st of 2020 due to the coronavirus. It's possible that the Environmental Protection Agency would have to declare a separate public health emergency under the Superfund law to comply with the statute. But Biden's EPA administrator could easily do so. Individuals eligible for Medicare would have to have spent six months in the geographic area subject to the emergency. Since the area is the entire United States, this should not be a problem either. So there's more to it, but in essence, that's the crux of the legal reasoning. Now, again, this is going to get challenged if Biden were to ever do something like this, and he wouldn't. But if you actually care about Americans who are suffering, who don't have health care, it's worth the fight. But the point is that Marianne Williamson is talking about the possibilities, things that Biden can do, because then that plants the question in that brains of MSNBC viewers. Wait, he can do this, but he won't. Why? Why won't he do this? This is really, really important because MSNBC's audience, Democratic Party's most loyal base, they are conditioned to unequivocally accept any and everything that the Democratic Party says and does because the alternative is so awful, the Republican Party, and they are crazy. But that's why Marianne Williamson also makes the point that if you want to stop Republicans from winning, you have to relieve people's pain because they're not gonna come out in standard line for hours at a time if they don't feel as if Democrats are delivering. It's just too high of a cost for not a big enough payoff. So this is very important because what Marianne Williamson is saying is correct and she's not gonna get through to 100% of MSNBC's audience but perhaps 10% just think for the first time, you know what, that's right. Maybe I should expect more from Democrats. Maybe I should opt for more left-wing Democrats since time after time, moderate Democrats have failed us after a Democratic Party administration, we get a more radical and extreme version of the Republican Party and maybe their failures is feeding into the radicalization because people are desperate and they're looking for answers and they're susceptible to being radicalized by a demagogue like Donald Trump or Ronda Santis or some other dickhead. So this is really invaluable what Marianne Williamson is saying here. Now one more clip that I wanna show you is her calling out Biden for being weak. This was incredible, just right off the bat. She tells Democrats, stop being wusses and I love this so much, I can't even tell you but let's watch. So I wanted to get to a few more points really quickly before we run out of time but you've also argued that Democrats like Joe Biden are missing their moment because they're too scared of being tarred as socialists and you write that like Biden, FDR got called a socialist and to destroy our capitalism and heard all the same things they say about progressives today. The difference however, Roosevelt didn't care. How would you advise Democrats to break that cycle today?  You know what Roosevelt said? I welcome their hatred. That's what he said. You know, Trump was a strong man. He was a political strong man, he abused power. That we do not want but neither do we need a weak president right now. We need someone as strong as Roosevelt and it's obviously apparently not enough to just have his picture up in the Oval Office. Roosevelt would be acting. Roosevelt would be taking what he called the drastic measures necessary to relieve people of despair and chronic tension and anxiety in terms of their health care in terms of the economy that is building up into an unsustainable rage. And if the Democrats do not respond to this we will see dark days ahead. She's exactly right. See the problem with the Democratic Party is that there's just no spark. There's no sign of life even right now in the White House. Biden has, you know, how many months, 10 months until the midterms he can do so much to try to relieve people's suffering just with executive order. He can call on Congress to send him bills even if he knows that it's not gonna get through mansion and cinema. He can at least put up the appearance that he cares that he's willing to fight but he won't even do the bare minimum. He won't even do political theater. And so this is a failure on his part. They're weak, they're wussies. And now, as she said, we don't need an authoritarian strongman like Donald Trump but we need someone who is going to vociferously advocate for what the American people needs. And what Democrats in Congress don't understand is that right now we have a limited window of time to act. And when I say we really, I mean them elected Democrats you have to do voting rights reform to stop a coup from Republicans. So you have to try to ameliorate some of the suffering from Americans if you want them to turn out to vote. Republicans are explicitly authoritarian at this point. They're against democracy and you don't have much room to govern if we lose more of our democracy. It's already hanging on by a thread. So at a minimum, you gotta at least try. You might fail but just showing us that you wanna fight is important in and of itself. But what Marianne Williamson is saying is that Democrats don't wanna do that. Either they just are uninterested in governing or they're too scared. Now FDR, he did say I welcome their hatred. He was talking to corporate America, talking to Wall Street and imagine a modern day Democrat doing that. It's just inconceivable. And the reason why this is the case, cynical take here is because the Democratic Party doesn't actually want to govern. They like when the Republican Party shifts further and further to the right because they're hoping to pick up these moderates that the Republican Party in theory is losing as they shift further to the right to appease their more fringe base. But the problem is that moderates aren't going to the Democratic Party. They're shifting to the right with the Republican Party. And as Democrats try to desperately court these moderates that they think Republicans are losing, they're losing their own base. They're losing young people. They're losing the left. But you see, it's a value proposition to Democrats. If you had the choice to have moderates in your base or the left in your base and you're sold out, you're completely beholden to corporate interests and Wall Street, who do you prefer? Well, of course moderates because with moderates there's no expectation there for you to deliver. You know, you can do tax cuts for the rich. You can try to further privatize Medicare and Social Security and they're not going to complain but the left is going to have an expectation. So the best thing that these corporate Democrats can do and hope for is to just bring in as many moderates as possible and push out the left.  That means that they include Republicans. That's the subtext. So, you know, either way, the Democratic Party is in very bad shape and what Marianne Williamson is giving them is common sense advice. Now Democrats who watch this, who are elected officials, they're not going to care. But what really is important is that MSNBC's audience listens to what Marianne Williamson has to say. And if some of them are receptive to her message, then that just is, you know, the small spark that might ignite the fire and make them realize the Democratic Party sucks. We need to opt for someone who's actually a left winger and not these milk toast moderates who continue to fail time and again.


Yng Ngôl ystod here at Rock Steel, what are your thoughts on the game? I think it was a really tough game. It was a really tough game. I think first half we dominated the game, we controlled the game, it just was missing probably that goal in that first half and obviously second half we knew they were going to push on there at home so they were playing the counter but I think the whole team, as a team we did really well with the counter, with the chance they were trying to create and we take the pointy from the game and that's the clinch here and the point. It seemed quite a stop-start with players getting injured throughout for Rosanne, for them a lot of changes and stuff like that. Was it difficult to keep the momentum going when they were on top and let the game flow? Yeah, definitely. When we got momentum and the ball was coming down, it was strange and it was cold as well so it made the game a little bit more difficult but what I'm going to take from the team is that we kept the concentration for the whole 90 minutes and we managed to get that clinch here. I think it's a thing because they were playing really deep, it was difficult for us to find the gap to go and show everything we want, the chance we were getting from us from outside the box but obviously we do stay, obviously I haven't kept a clinch here in a while so for us we all kept a clinch here away from home, it was really important for us. They did have some decent moments so to keep a clinch here I guess shows that it was a good defensive performance from us. Yeah definitely, obviously if you keep clinch here you're going to end up winning games so that's that attitude we have to take on from this game to keep as many clinch here as we can and obviously give as many opportunities as we can to our forward players to win us the game.  How did you find it today? Is anything different for you in terms of what the gaffer is asking of you fundamentally? No no no different, I feel comfortable playing in the midfield, playing at the back like I said before, I would play in any position the gaffer won me, I was going to give like 110% and today playing alongside Pordy he always makes it easy for me because obviously he's got experience in this league and we've got a good connection so it's not a problem where the position I play in the team. So over there, that stand behind you, 2500s of gravity supporters, more than home supporters in the end I guess coming out and seeing that on and away game like this gives she was plays a massive boost. Yeah definitely, you know, like these fans have been standing amazing all season with us, you know, like obviously we didn't really but I think expected it after you know the bad performance we give at Carlabne especially in the second half you know that they would turn up like this but I think you know the game against Hofford give us like something to believe in and obviously today I think we've got to see that. I think they really help us to get that clean sheet today and get that point at Rojda. Is it positive to take into Tuesday night especially, you know, values that we've shown and keeping the ball out of the back of our net, positivity to take forward? Yeah definitely, it's all positive to take, it's obviously to take the clean sheet, the team that we defend as a team, we attack as a team, you know and it's like positive as well in fact the gaffer got the option to mix the team around and still get our performance so it's loads of positive to take from this game and hopefully we can show that again. I'll see you on Tuesday night.


That's right son. That's a horse's head a thoroughbred horse and that black bar there means first First in name and first in battle. There's only one outfit that fits that description. It's the first cavalry division a Horse meet did you follow his ride from on horseback and charge the enemy? Well, that was the idea up to the last war one of our regiments fought on horseback in the Civil War The cavalry were the tanks and reconnaissance of those days The end of the Civil War didn't end the fight for the cavalry They had a job to do out West fight ninjas They got down from their horses pretty often to do the actual shooting like General Custer in his seventh regiment at the battle of the little big horn That seventh regiment of Custer's was the granddaddy of the first cavalry division, which was formed in 1921 They kept operating with horses until they went to the South Pacific in 1943 as dough foot Cavalry sabers were turned in for bayonets and each GI did his own steeple chasing. I Guess they figured the jungle was too rough for the tender skin of a horse And old brother so was that combat training. We got to know what the nickname tender foot really meant tender feet Wet feet to when we practiced amphibious landings in February 1944 we went gunning for the Japs on Los Negros Island in the Admiralty's Was a good thing we didn't have our mouth so long Racket those Navy guns made would have scared them to death like it almost did us We loaded the landing craft the same way we did on maneuvers But it wasn't a rehearsal this time It was a public performance Our throats were drier than they've been even after a hard ride on a dusty trail Got a sure okay looked easy But our troubles was just beginning The Japs had Imperial Marines to protect their airstrip They was cracked troops big fellas like the little Mavericks would expect it Half of them was local drunk They charge us singing deep in the heart of Texas honest to God Didn't fool us none Ain't her to jab with the Texas draw again It was our real wildcat scram But tough as they were we fought them at point blank range sometimes man-to-man We had a lot of heroes in that scrap but not all lived to get their medals Including this signal core cameraman who made his last picture that day. It's the same we beat the Japs hollow Killed off our big herd of them There's a way their airstrip in their islands The chief was pleased Now every man tank and gun to point over the water to the Philippines to batang The first cavalry was the first to hit Leyte in November 1944 We rode some strange beasts into the Landon Amtrax Buffaloe Alligators LCVP Rough riding to Didn't get saddle sore, but almost got seasick Made our stomach steadier though to see the rocket ships And they are forced given the Japs are going on They called the place we came ashore white beach Wasn't quite so white when we left it We were pinned down Some of our boys caught Japs bullets. It would have been worse though without that mechanized stuff They gave us for horses Pretty tough hide on those Shermans and Amtrax tougher than the ponies tougher even than the dough puts Leyte was no breeze. It was hard fighting We had to smoke the enemy out of their holes like coyotes But we smoked them out. All right The supply boys did great work hustling stuff to the fire We kept blast madam with rifle fire bazookas Rifle grenades So we put the first cavalry brand on every jet Maverick Then we moved into Teclobon Ladies number one city might be glad to see us Us and those good old American smokes we carried. There wasn't time for a long break though In January we were on Luzon driving hell-bent for the Capitol in Manila Driving a hundred and fourteen miles in sixty hours stopping only to smash resistance But we did it We beat every other outfit in the Manila not just for the glory But to avenge one of our own troopers An officer who had commanded the brigade of the first cavalry back in 1939 before he went to the Philippines in Corregidor The man whose name was Jonathan M. Wayne might our next APO was Tokyo We figured in Vedin, Japan would be plenty rough But a little something called the atomic bomb made it possible to enter Tokyo riding in a parade instead of fighting our way in First to get here And figure any other outfit would make it before us. Did you smart-looking troopers aren't they mind you? I said troopers not soldiers We did this job without horses. All right, but we're still troopers troopers of the first cavalry division


What we like about the attendees here is it's a very technical audience. It allows some of the development community in the Linux space to come talk to our developers. The Linux Foundation events tend to draw the people who are actually working on open source projects. So I get a chance to meet with people who are also working in upstream communities and people who are actually influencing the way open source is developed.


I'm Elizabeth Zeta with Caltrans. If a traffic light is out, treat it as a stop sign and proceed with caution.


In the previous lecture we looked at how to set up governing equations of motion for using Hamilton's principle and we defined a quantity known as Lagrangian which is the difference of kinetic energy and potential energy and we defined an integral known as action integral and according to Hamilton's principle motion takes place in such a way that this action integral is rendered minimum. So a condition for stationarity of this action integral is obtained through this equation or obtained as this equation and this is known as the Lagrange's equation. Now we applied this principle to simple dynamical systems like 2 degree of freedom, discrete multi degree of freedom system. We set up the expression for the Lagrangian in terms of kinetic energy and the potential energy and we showed that that approach leads to a pair of second order ordinary differential equations which are coupled to each other. Now this pair of equations could as well be obtained by a drawing free body diagrams of M1 and M2 and using the Alamed's principle but this is a different approach where we deal with only scalar quantities like energies not the vector quantities that we will be using if we were to use the Alamed's principle and draw free body diagrams and so on and so forth. Now what we will do we also covered the case of an axially vibrating rod how the Hamilton's principle can be applied. So here the action integral is obtained as an integral over the domain of the rod that is integral 0 to L Mu dot square DX and this is the strain energy and this is the action integral and by using the various calculus of variations we obtained this as a field equations and I also pointed out that the variational approach also leads to the definition of the appropriate boundary conditions and the four generic you know axially vibrating rod problems where the two ends are free, one end is fixed and other end is free and both ends are fixed are obtained as I mean the appropriate boundary conditions for these four cases are obtained as a part of this solution. Now what we will do in today's class is we will extend this example the approach to a few examples on axially vibrating rods and then we will apply the method to the problem of flexural vibration of beams. So the few examples that now I will consider helps you to analyze apart from setting up the field equations we will also obtain the appropriate boundary conditions, the emphasis is on obtaining the boundary condition. So here we have an axially vibrating rod which is mounted on a spring here and at this end it carries a mass through a spring as shown here. Now the kinetic energy of this system consists of kinetic energy stored in this bar and the kinetic energy of this mass that is written here U of X, T is the axial displacement of this bar and V is the displacement of this mass, so we get the kinetic energy as shown here. The potential energy is a contribution from strain energy stored here and the energy stored here and energy stored here and that is written in this form. Now the action integral is obtained in terms of the energy stored in the various elements and that is as shown here. Now in this example I will run through the various steps again, so we will assume that U of X, T and V of T are the unknown optimal solutions and U bar of X, T and V bar of T are the class of admissible solutions which are obtained by adding the variation, one variation eta 1 X, T other one eta 2 of T to U and V respectively. So now the action integral is now parameterized in terms of the two parameters epsilon 1 and epsilon 2 and we get this equation. Now we know that the action integral would reach the stationary value, the necessary condition for that is the gradient with respect to epsilon 1 must vanish and gradient with respect to epsilon 2 must vanish and we know beforehand that this stationarity condition will be realized when epsilon 1 is 0 and epsilon 2 is 0 and similarly epsilon 1 and 2 are 0 because by definition U and V are the unknown optimal solutions. Now if I implement these two requirements we will get these two equations and here we have now terms involving time derivative of the variation and special derivative of the variation the time derivative appears here as well as here in the second equation. So we will deal with these three integrals first and then come to the other part of the equation. Suppose if you now consider T1 to T2 Mu dot eta 1 dot dt I will integrate by parts and I will get the first term is this and the second term is this. Similarly on terms involving special derivative of the variation we again integrate by parts this is the first term I get this is the second term. The second equation for the variable V dot of T there is a eta 2 dot of T and integration by parts now leads to these two terms. Now eta 1 X comma T and eta 2 of T are arbitrary variations and eta 2 of T is 0 at T1 and T2 so what happens is these two of these terms go to 0, see these terms are appearing here and here these two are 0. Now we need to deal with the other terms now reverting back to the original equation I get now the statement for stationarity of A as this equation and we have to now collect terms involving eta 1 of X comma T which is done here eta 1 of 0 comma T and eta 1 L comma T and if you organize these terms we will get the field equation since eta 1 X comma T is arbitrary, eta 1 L comma T is arbitrary, eta 1 0 comma T is arbitrary the only way this sum can be 0 for all eta is that individual terms must be 0 and using that condition I get this as a field equation and these two as the appropriate boundary conditions. So these boundary conditions for example at X equal to L now involve the force transferred through this spring and X equal to 0 the force transferred through K1 as you can see here. Now similarly for the point mass if we use the second equation we get the governing equation to be this, this we have seen earlier for a simple single degree freedom system. So the final equations these two are the field equations, these two are the boundary conditions and we have to specify the appropriate initial conditions to solve the problem. So the point that we can take from this exercise is that the variational method helps you to correctly identify the boundary conditions along with the derivation of the field equation. So I will run through a few more examples I will not work out all the steps in detail but I will mention, so this is once example where a spring K connects the two ends of the rod. So now strain energy will be given by strain energy stored in the bar and the energy stored in the spring as shown here this is the strain energy stored in the bar and this is the energy stored in the spring this is the kinetic energy this is only the energy stored in the bar. So this is the Lagrangian and if we apply now the variational argument we get the field equation to be this, this remains same as what we got earlier but the boundary conditions now at X equal to 0 we will have this additional term which is a force transferred through this spring at this end and similarly at X equal to L I will get another force which is a force transferred at this end through the spring. Now how about an inertial element being present at the boundary, suppose if I have now a point mass at the end and again now the kinetic energy will consist of the energy stored in the bar plus the kinetic energy of this mass which is given here and the strain energy will be the energy stored in the bar, so this is this expression. So if you apply the Hamilton's principle we will get this as a field equation, now at this end the bar is fixed this is the boundary condition, at the other end now the axial thrust due to elastic action has now there is an additional term which is due to the inertia of this mass, so that will appear in the boundary condition. So along with this of course we need to specify the appropriate initial conditions. Now slightly more involved example now there is a there are two springs and another inertial element this is an exercise that you need to do the hints are given here this is the expression for the kinetic energy, this is the expression for the potential energy and I have worked out few steps if you work through this you will get the field equation, the equation of motion for the discrete mass that is this is given by this and for the bar this is this and the boundary conditions as you can expect there will be a term involving the force transferred through the K1 at this end and a term involving force transferred through K2 at this end, so that is what you see at L equal to T there is a force through K2 and at L equal to 0 there is a force through K1. Now this logic can be extended now to other structural elements like Euler Bernoulli beam plates and plane stress elements and so on and so forth, so for purpose of illustration I will consider one example that is case of a Euler Bernoulli beam, so this beam has span L, flexural rigidity EI of X and mass per unit length M of X. The kinetic energy is given by this expression M of X into V dot square DX I am considering the underlying beam theory is the Euler Bernoulli beam theory therefore there is no contribution to kinetic energy from rotary inertia, similarly when I write potential energy this is the strain energy due to bending and there is no contribution to this energy from shear deformation, so if you include rotary inertia and shear deformation we will get the corresponding formulation for a T-Moschenko beam, but right now we will focus on Euler Bernoulli beam and this is the Lagrangian T-V and this is a action integral, so we can see that the Lagrangian is now a function of V double prime that appears here and V dot which appears here, so I can write action integral in terms of this function as F of V double prime V dot. Now as before we consider this to be the unknown optimal solution and this is the variation and this is the class of admissible function which have now been parameterized in terms of epsilon, so A as a function of epsilon is this, now we know that optimal value of A as a function of epsilon is reached when epsilon equal to 0, so we apply this condition, so DA by D epsilon is dou F by dou epsilon which will be dou F by dou V double prime into eta double prime plus dou F by dou V dot into eta dot, so the bar we are not maintaining distinction between V bar and V because this calculation is being done at epsilon equal to 0 and in which case V bar and V are identical, so this is the equation. Now we again consider these two terms separately, so we have 0 to L dou F by dou V double prime eta double prime X comma T, so one integration by parts will lead to this equation and on the second equation we will again perform another integration by parts, so I will end up with these three terms. Similarly if you consider this term and consider integration from T1 to T2 we get again integration by parts we get these two, now at T equal to T1 and T2 the variation is 0, so this term drops off, so we have to still interpret the other terms which will still remain till we complete their interpretation, so we carry them further in our derivation and we get this is now the equation which is a condition for stationarity of A. Now we'll again use the arguments that variation is arbitrary, therefore the only way this term can be equal to 0 is when each of these terms are individually equal to 0, so since eta of X comma T is arbitrary the term inside this brace must be equal to 0 I get this and similarly term inside the bracket must be 0 and term inside this bracket must be equal, so now this is a still that F is remaining, the notation for F is, F is this, so if you now insert all this calculation you know dou F by dou V double prime dou F by dou V dot etcetera in this we will get the field equation which is the well-known Euler-Bernoulli beam equation, this is a field equation, now the boundary conditions are to be derived from these conditions. Now at free end when displacements are not specified we will not be able to assign any value to the variation therefore the term that multiplies that must be equal to 0, so at a free end neither displacement can be specified nor slope can be specified, therefore for this term to be equal to 0 it is natural that this multiplier must be equal to 0, so on the other hand if at the X equal to 0 or L if displacements are specified, for example in a simply supported beam the translation is specified to be 0 in which case this will be 0, and similarly at the fixed end the slope will be 0 therefore this will be 0, the variation would be 0, so there are now 16 combinations of one span beams that are possible and I have just displayed them here, this is both ends are free, this is one end is fixed other end free and so on and so forth. So now we can examine suppose we are focusing on one of these boundaries and I look at this boundary and consider these expressions we can see here that the slope cannot be specified therefore EI V double prime must be equal to 0, we know that EI V double prime represents the bending moment, bending moment is 0, and similarly the shear force is 0 which comes from this equation because displacement cannot be specified, so this is 0, at the clamped end we know that the displacement is 0 therefore the boundary conditions emerge as V prime equal to 0, V equal to 0 which again satisfies this equation because the variation must satisfy these conditions. Similarly it is simply supported end, the bending moment is 0 and the displacement is 0, at this roller end the slope is 0 and shear force is 0, so we can therefore see that consistent with the variational formulation we can think of 16 possible single span beams and we can set up all the appropriate boundary conditions. Now slightly more involved example, suppose an Euler Bernoulli beam is resting on a Winkler's foundation and it also carries an axial load P, now what is the kinetic energy? The kinetic energy will be due to flexure of the beam and this is given by this, the potential energy now will be due to the flexure of the beam and due to the elastic foundation and contribution from axial loads, so if we write these expressions this is flexure, this is from Winkler foundation and this is due to the axial compression load. So we can form the Lagrangian and again construct the action integral, now the F function that is the Lagrangian now has because of Winkler foundation and the axial loads we have now in addition to V double prime and V prime we also have V and V prime, so we can do this, we get this as the Euler Lagrange equation and the field equation turns out to be this and the boundary conditions now will involve for example at the two ends the axial thrust will appear in the boundary conditions and the bending moment would be 0 at the free ends, this is the shear force at the two ends, so I leave the intermediate steps for you to work out. So now the beam carrying some inertial elements and a spring this again helps you to appreciate different facets of this formulation, so the kinetic energy now has contribution from this mass, contribution from this beam and from this mass M2, M1 and M2 are point masses, so this is the expression for the kinetic energy, U of T is the displacement of this mass and V of X, T is the displacement of the beam. The potential energy now has the one that is stored in the beam and the one that is there in the, this spring and we get this, so now you apply the variational formulation you will get one equation for U, another equation for V and the equation for V reads as shown here and the equation for U is this and the boundary conditions at X equal to 0 we can see that bending moment emerges to be 0, but the shear force will have contributions from at this end from this mass and at this end there will be a shear force contribution from this spring, so these are the appropriate boundary conditions. Now how about a beam with a overhang, so there are now three places where we need to worry about the boundary conditions, so what I do is I call displacement field in this portion as V1 and in this portion as V2 and we will assume that EI and MR functions of X as X varies from 0 to L1 plus L2, so the kinetic energy now has contribution from this segment of the beam 0 to L1 and L1 to L1 plus L2, similarly the strain energy flexure in this part plus flexure in this part, the action integral is now therefore T minus V is this expression, so now I will take V1 to be a class of admissible, parameterized admissible set of functions is formed by for both V1 and V2 as shown here, okay. Now so this is the action integral in terms of epsilon 1 and epsilon 2 and these are the conditions for stationarity of this, so if you run through this calculation you will have to do this differentiation and the corresponding integration by parts and if you carry out these calculations we get the field equations for X varying from 0 to L1 and X varying from L1 to L1 plus L2 separately and they are shown here, and now the boundary conditions you will have to consider what happens here, what happens here and what happens here, so the required conditions of compatibility and equilibrium of forces etc is automatically satisfied in this formulation and we get, I mean you can go through these steps, these are simple manipulations and we get the boundary conditions as shown here, so this is at X equal to 0, bending moment would be 0 and displacement would be 0, here the bending moment would be 0, shear force would be 0, here what happens, so at X equal to 0 these are the boundary conditions at the X equal to L1 plus L2 that is the right hand free and this is the bending moment and shear force are 0, at the intermediate point this is the equilibrium of moments and this is compatibility and V1 and V2 must be equal to 0, right, so we have now 8 boundary conditions and 2 4th order equations and along with that we should specify the required for initial condition, so this gives a formulation for a beam with overhang, so I suggest 2 exercises you can consider a proper cantilever beam with a hinge in between, you now formulate the you know the equation of motion and obtain the appropriate boundary conditions, the other example is a beam that is spinning about this Y axis, okay, so you have to correctly write the expression for kinetic energy and the energy due to bending and derive the appropriate field equations and boundary conditions. Now at this stage it is useful to quickly review the solution of equation of motion for discrete multi degree freedom systems, so this is a generic form of equation of motion for linear systems, time invariant linear systems MX double dot plus CX dot plus KX equal to F of T, X is displacement vector, M is mass matrix, C is damping matrix, K is the stiffness matrix and these are the initial conditions. Now what I will, I don't intend to do this review as a part of this course because this review is already available in another NPTEL course which I have taught and this is the website where this is available and if you go to lecture 13 slides 3 to 50 and lecture 14 slides 2 to 24 you will get this review, so I will assume that when you reach this point you will go to these lectures and familiarize yourself with the contents of this lecture before proceeding further as a part of this course. In the same course that I mentioned in lectures 15 and 16 there is a review of solution of equation of motion for continuous multi degree freedom systems specifically in the context of an Euler-Bernoulli damped Euler-Bernoulli beam and all the steps involved are again completely outlined here they run over 2 lectures, lectures 17, 15 and 16 and these are the slide numbers. So again I presume that before you proceed further in this course you will visit these 2 lectures and familiarize yourself with the contents of those lectures. Now the next topic that I would like to discuss is known as Rayleigh's quotient. Now the strategy that we are using right now is we want to first outline what are the approximate methods for analyzing vibrating systems and we will outline a few methods and that will lead to that will provide the motivation for us to understand how the finite element method is developed. So as a prelude to that we will start with what is known as Rayleigh's quotient. Now if you consider a n degree of freedom system with mass matrix M and stiffness matrix K where M is symmetric and positive definite, K is symmetric and positive semi-definite we will denote by omega I and phi I the ith eigen pair, so these are obtained by solving the eigenvalue problem associated with these 2 matrices and phi is the modal matrix, so it is the matrix of the eigenvectors, so this is a n cross n matrix each column represents one eigenvector and I assume that these eigenvectors have been normalized so that phi transpose M phi is an identity matrix that means these normal modes are mass normalized and phi transpose K phi is a diagonal matrix where the diagonal entry is omega I square, okay, so these are the orthogonality relations and the normalization conditions used. So if you take now the nth eigenvalue and nth eigenvector the equation that it satisfies is given by this, now if you pre-multiply by phi n transpose and on both sides I get this equation and K is a n by n matrix, phi n is a n cross 1 matrix so this will be a scalar because this is 1 cross n, n cross n, n cross 1 therefore this is a scalar so I can divide this expression for omega n square as phi n transpose K phi n divided by phi n transpose M phi n, so this is an exact result, okay. Now instead of considering this nth eigenvector that eigenvector I will now replace by an arbitrary n cross 1 vector I form the ratio U transpose KU divided by U transpose MU, but definition I call this quantity as Rayleigh's quotient, so clearly we denote it as R of U, U is a vector and it has clearly the units of radian per second whole square. Now if U were to coincide with the nth eigenvector R of U would be simply the nth natural frequency, okay. Now why we are interested in this quantity? This quantity has two important properties namely this is bounded between the first natural frequency the first eigenvalue and the last eigenvalue, and also the second property says if U is in the neighborhood of phi n, if U is phi n plus epsilon Y where Y is a vector which is a perturbation R of U which is the eigenvalue will be in the neighborhood of omega n square but this correction is order epsilon square. The error in the vector U, the error I mean the departure from the true eigenvector if it is order epsilon, the error in the eigenvalue will be of the order epsilon square, so this helps us to evaluate the eigenvalues in approximate manner by making guesses on the eigenvector. So if you make an error of order epsilon in making the guess you will end up with an order epsilon square in your estimate of the eigenvalue, so this is an advantage. So we will see this with a few examples and also I will now show why those two statements are true. Actually the main application of Rayleigh's quotient is that it can be used to approximate the first natural frequency of the systems by making guesses on the mode shape that means we are going to use this part of the result, so if my objective is to estimate the first natural frequency I will start by making a guess on the first mode shape and substitute that in this expression and I will get an estimate for the first natural frequency, okay. Now we will see that through an example, suppose if you consider now a 3 degree freedom system, these are the parameters you can easily write the equation of motion, so this is the equation of motion, this is the mass matrix, this is the stiffness matrix, and if you carry out the eigenvalue analysis you will see that the modal vector is, modal matrix is given by this, this is the diagonal matrix of squares of natural frequencies and this mass, this modal matrix is mass normalized you can verify that, and the phi transpose K phi is nothing but this lambda, so this is an exact result. Now if I now make a guess U for example 111, what will be R of U? You can compute that, that means I am assuming actual mode shape if for example is 1.13, 1.63, 1.78, so what I will do is I will take that as 111, so let us say guess on my mode shape, then my approximation to the R of U turns out to be this number 22.22, so this is if you are interested in estimating the first natural frequency this will be the your estimate of the square of the first natural frequency, if you take something like 1, 2, 3, right this will be the estimate, on the other hand if you take say 1.1, 1.62 you come closer to the true natural frequency, now if you take 1, minus 1, minus 1 this will be a bad guess on the mode shape if you are familiar with how mode shapes behave this will not be a good guess on the first mode shape therefore this is depicted in the Rayleigh's quotient, but no matter what you are doing we see that Rayleigh's quotient is bounded between the first natural frequency square and this last natural frequency square in all these examples. Now among these various trials on your guesses on the first mode shape this one gives the lowest Rayleigh's quotient, so if you actually in this case I know 17.25 is the answer, but if you are applying this method under a situation where you do not know the exact answer you should accept this as the best possible approximation to the square of the natural frequency because of this bounding property, or if you can at best be only equal to the first natural frequency square it cannot drop below that, so because of that bounding property the lowest among this is the best possible estimate for the first eigenvalue.  Now if I make this as a initial guess to maintain this as a normal mode for the system I should supply additional forces which bring back this point here, brings back this point here, this point to this you will be supplying additional forces to the system and that does additional work and that contributes the potential energy, so therefore the natural frequencies go up, so to maintain the approximate mode shape we need to apply additional forces and this leads to increase in the potential energy of the system which in turn increases the natural frequency, that explains why the Rayleigh's quotient provides the bound. Now we can quickly argue out why these results are true, so I will start with the first result that R of U is bounded between first natural frequency and the highest natural frequency, how do I show that? Now U is a vector N cross 1 vector therefore it can be expanded in terms of these true mode shapes, so I can write U as a linear combination of these mode shapes where N takes value from 1 to capital N, now using this R of U can be written in this form, now since phi N's are true mode shapes and they are orthogonal with respect to K and M okay, and they are mass normalized if you expand this out all the, this will become a double summation but all the cross terms will go to 0 and I will be left with this in the denominator, similarly in the numerator the diagonal terms will have omega N square multiplying alpha N square, so this will be the Rayleigh's quotient in terms of the unknown true natural frequencies and these unknown alpha N's. Now I will manipulate this, what I will do is I will expand this and pull out omega 1 square outside, so first term will be alpha 1 square as it is omega N square has gone outside and this is other terms are this and the denominator remains the same. Now there is an ordering of natural frequencies, omega 1 is less than or equal to omega 2 less than or equal to omega 3 and so on and so forth that is how the labels on eigenvalues appear, so consequently we can see that these terms inside the bracket here will be greater than or equal to 1, so consequently we see that this inequality holds, okay. Now to get the other bound what I will do is now I will pull out the last frequency outside and divide by omega N square all other terms, so this will be, this multiplier will be 1, now by the same logic now all these numbers inside this bracket will be less than or equal to 1, so consequently this bound emerges, so we get this the result that Raleigh's quotient is bounded between first eigenvalue and the last eigenvalue. Now we can work through this if the U that we are selecting is orthogonal to the first S modes, it so happens that you know first three modes and you select the, I mean in a 10 degree freedom system for example if you select your trial function to be orthogonal to the first three modes then you will end up getting an upper bound on the fourth natural frequency, so that is the third natural frequency, that is this result can be shown using similar arguments I leave this as an exercise for you to complete. The second result that I stated has this notation order epsilon square, okay, now what I want to do is first I want to define what exactly is this notation means, so we will see that first then before going to the proof, so we use what are known as the capital O and lower case O notations, now if we consider a function F of X and if we say that it is order G of X as X goes to A, what it means is if X of A is in the neighborhood, X is in the neighborhood of A by this distance delta, then this F of X will be less than or equal to this M into G of X where for every A there exists a M greater than 0 and delta greater than 0 so that this inequality is whole square, operationally to verify that what we need to do is we have to divide F of X by G of X and take the limit extending to A, if this is finite then we say that F of X is order G of X as X goes to A. The lower case order G of X means this actually goes to 0, it is not just finite but it actually goes to 0, okay. Now some few examples, for example we have this A X to the power of 7 plus B X Q plus C X plus D, now the question is is it order X to the power of 7 as X goes to infinity, how do I verify? I will form this ratio and you can see that this first term will be A and all this will be 1 by X to the power of 4, etc. as X tends to infinity this goes to A, it is finite therefore this is true. Now is this order X to the power of 0 as X goes to 0? Again we see this as X goes to 0 all these terms will drop off there is a D here which is finite therefore this is also true. So A X to the power of 7 is order X to the power of 7 by this logic again, whereas A X to the power of 7 is not order X to the power of 8 because this will be 1 by X and as X goes to 0 this becomes unbounded. A few more examples, sine X is order X as X goes to 0 because sine X by X as X goes to 0 is 1 which is finite, similarly sine X square is order X square because of this identity. So there are a few more examples you can verify and this order notation will be using soon in this part of the lecture, but later on when I come to numerical integration of equilibrium equations I will be using these notations again, so that's why I have introduced at this stage. Now what is the result that we are talking about? If U is in the neighborhood of the true nth eigenvector and the departure is epsilon Y that is order epsilon Y, then the Rayleigh's quotient will be omega n square plus order epsilon square, so how do I show that? So you can again write R of phi n plus epsilon Y you write this and you expand this and phi n is a true mode shape therefore it will satisfy the, I know this is a square of the natural frequency and this is 1, so that's what I write and I am left with these two terms. Now this Y is the perturbation, now I will take Y to be in terms of all the modes except the nth mode that is how we can write this. So if you now substitute this we can show that phi n transpose K Y is 0 because every, we are excluding I not equal to n that is 0 and phi n transpose MY is 0 and this is 0 and we get Y transpose Y as this which is this is here and Y transpose K Y, Y transpose MY is here, Y transpose K Y is here, the term I not equal to n in this summation I cannot take value of n. Now if you go back and substitute you get this ratio and the denominator I can write in this form and if I do a binomial expansion of this and retain only the first few terms I get this expression and we can see that the Rayleigh's quotient is possesses the property that I stated, that means this term is order epsilon square, okay. So if you make a reasonably good guess on eigenvector you will get reasonably good much better answer on eigenvalue that is the story. So this also means that this Rayleigh's quotient has a stationary value in the neighborhood of an eigenvector, it indeed takes the minimum value in the neighborhood of the fundamental mode, this result this is known as Rayleigh's principle, okay. Now this will be useful in some of our formulations. Now the discussion on Rayleigh's quotient can also be based on a slightly different argument and that is based on energies, so if I now consider for purpose of illustration I will consider a simple system, a mass spring system there is no energy dissipation here therefore it's a conservative system, the total energy is conserved, so this is an expression for kinetic energy, this is an expression for potential energy, if I add these two energies I get the total energy which must be a constant because there is no energy dissipation in the system. Now in normal mode oscillations we are interested in harmonic oscillations, so in a multi degree freedom system we say that a system is executing normal mode oscillations if all points on the structure are vibrating harmonically at the same frequency, so in this case it's a very trivial case, so if it is executing harmonic motions U of T will be given by this. Now let us substitute this into the expression for kinetic energy, I get kinetic energy to be MeA square omega square by 2 sin omega T, sin square omega T and this is cos square omega T minus alpha. Now total energy is sum of this, if you add now this sin square and cos square will add up and I get this is a constant that I am talking here, now this would automatically means that when kinetic energy reaches its maximum value the potential energy must be 0, and similarly when potential energy reaches its maximum value the kinetic energy will be 0, so if we write that you can see here the expression for kinetic energy has sin square omega T minus alpha, the potential energy has cos square omega T minus alpha, the maximum value that functions sin square omega T minus alpha can take is 1, similarly cos square omega T minus alpha the maximum value that this function can take is also 1, so I get the maximum kinetic energy to be this, maximum potential energy to be this and since they are equal I get the expression for the natural frequency, so I have been able to write this expression for the natural frequency without writing the equation of motion, I am simply using the fact that I am looking for harmonic oscillations, harmonic oscillatory responses because that is how we define the normal mode oscillations. Now how about the similar argument for a multi degree freedom system, so if U dot is the velocity vector, M is the mass matrix, we can write the kinetic energy as half U dot transpose MU dot, and potential energy as half U transpose KU, the total energy will be the sum of these two and again this must be a constant, in normal mode oscillations all points on the structure vibrate harmonically at the same frequency, this is the amplitude of those oscillations and this is the frequency and this phase difference could be either 0 or pi in an undamped system, that means it is a synchronous motion, all points vibrate harmonically at the same frequency so that the ratio of oscillations at any two points is independent of time, that also means all points reach their respective maximum values simultaneously, maximum and minimum values simultaneously, that means phase is either 0 or pi, that means points can be perfectly in phase or perfectly out of phase, there is only two phase differences possible. Now for this type of motions which are the normal mode oscillations I can write the kinetic energy, this is given by this, potential energy is given by this, now the maximum value of kinetic energy is reached when sine square omega T minus alpha is 1 that gives me this, and maximum potential energy is reached when cos square omega T minus alpha is 1 that gives me this expression. So if I equate these two I get the expression for omega square which is the U transpose KU, U transpose MU and now you will recognize that this is nothing but the definition of Rayleigh's quotient for a trial vector U, clearly when U coincides with the true natural frequency of, true normal mode of the system this will be the corresponding natural frequency or the corresponding eigenvalue which is square of the natural frequency. Now how about for a continuous system? Now let me consider this axially vibrating bar, U of X, T is the axial displacement, L is the span, A is the axial rigidity, M of X is the mass per unit length, so we have derived these expressions for kinetic energy and potential energy which is strain energy, so again in normal mode oscillations all points on the structure vibrate harmonically at the same frequency, and phi of X is now the eigenfunction which is not known, okay. So now I will substitute this into the expression for kinetic energy, and again notice that the maximum value will be reached when cos square omega T minus alpha is 1, and sine square omega T minus alpha is 1 respectively for the potential energy and the kinetic energy, I will get maximum kinetic energy to be this, and maximum potential energy to be this. So these two need to be equal, so based on that I get now the expression for the frequency of oscillation, and this is defined as now the Rayleigh's quotient for axially vibrating bar. Now unlike in discrete multi degree freedom system, this Rayleigh's quotient here only satisfies the bound at this end, because the highest natural frequency of a distributed parameter system goes to infinity, there is no bound on this side, it virtually means the Rayleigh's quotient is less than infinity, but that is not worth stating, so we only get bound on one side. How about the Euler Bernoulli beam? Again we consider this beam system, L is the span, EI is the flexural rigidity, M of X is the mass per unit length, and displacement in the Y direction is V of X, T, so this is the expression for kinetic energy, this is the expression for the potential energy. Now in normal mode oscillations again we take the response to be phi of X cos omega T minus alpha, so this is the expression for the maximum kinetic energy, this is the expression for maximum potential energy, these two must be equal, so I get the definition of Rayleigh's quotient for a beam in terms of flexural rigidity and mass per unit length as shown here, again we notice that the Rayleigh's quotient is greater than or equal to omega 1 square. So in the next class what I will do is I will apply this idea of Rayleigh's quotient for a few class of systems and show how it is useful in characterizing the natural frequency of obtaining approximation to the fundamental mode of vibrating systems. Now the success of this quotient, Rayleigh's quotient as an approximation to the first eigenvalue depends on how well you can approximate the mode shape, so it is not always that you can make intelligent guesses, so you need to have a systematic procedure to lower the Rayleigh's quotient. Now how do we lower the Rayleigh's quotient, so to be able to do that we should introduce certain undetermined parameters in my definition of the trial function, this is the assumed mode shape, so if I assume phi of X as a series, if I do expand this unknown trial function in a set of say orthogonal functions with unknown undetermined coefficients then I can select those undetermined coefficients so that the Rayleigh's quotient is minimized, right, so that gives us a systematic way to lower the Rayleigh's quotient and therefore obtain better approximation to the first natural frequency. Now in doing so we not only get a better approximation to the first natural frequency but we will also end up getting approximations to the higher natural frequency, this method that I am talking about is known as Rayleigh Ritz method, so in the next part of the lecture after illustrating the applications of Rayleigh's quotient we will develop the Rayleigh Ritz method and then again apply to a few methods, I will conclude this lecture at this point.


Live from Las Vegas, it's theCUBE. Covering Dell Technologies World 2019. Brought to you by Dell Technologies and its ecosystem partners. Welcome back everyone, you are watching day three of theCUBE's live coverage of Dell Technologies World here in Sin City, Las Vegas, Nevada. I'm your host, Rebecca Knight, along with my co-host Stu Miniman. We have Varun Chabra, who is the Vice President, Product Marketing Cloud Dell EMC. Welcome back to theCUBE Varun and Mark Lohmeyer, SVP GM of Cloud Platform VMware. Thank you so much for coming on theCUBE. Thanks, great to be here. So before the cameras were rolling, we were talking that it should be a rap song, VMC on Dell EMC. Tell us about the news this week. Yeah, sure, so maybe I can kick it off. So really excited this Monday to introduce VMware Cloud on Dell EMC. And as I sort of think back to when we first started our discussions together between the two companies, we really had this sort of design goal in mind, which was how do we bring the simplicity, the agility, and sort of the consumption economics of a public cloud model, right? But with the control, the security, the enterprise class capabilities, performance that customers expect in an on-prem environment. And how could VMware and Dell work together to really jointly engineer something that we think would be really special and achieve those goals, right? And based on feedback we got from customers, we're really pleased to sort of the reaction to this. And we think that's really going to hit the sweet spot of kind of best of those both worlds. So Varun, Dell EMC's been in the private cloud market for a bit. Actually it was somebody on the EMC side that, as far as I know, was credited with coming up with that terminology. So some of this isn't new. Give us what is new about this offering compared to what we've done in the past. Yeah, great questions too. So essentially what is really innovative about this is that this is taking the public cloud model to on-premises, as Mark said. It's a fully managed service where Dell Technologies and VMware are working together behind the scenes to provide that public cloud-like experience, the hands-off operations, the ability to provision resources using a cloud portal. And have it be installed for you and it's set up. Once it's set up, software patches, operating system updates, hardware updates, all of them are basically going to be managed for you. If there's any support issues, the VMware team will file a ticket for you. You don't even need to file a ticket. It will be managed, but managed for you. This issue will be resolved. We think that this will be a really transformative way for customers to consume cloud resources. And this is all about bringing the cloud model to the data center where there's so much data that customers already have. All right, so Mark, there were ripples in the industry a couple years ago when the VMware cloud on AWS was out. Some people may be like, hey, why wasn't it done on the Dell stuff first? But the thing I want to ask is, what have you learned from that AWS engagement and how did that impact what you're doing now on the Dell EMC? Yeah, it's a great question. So I think one thing, so we learn from our customers and the feedback they give us. And one of the things that they shared is, look, they really like the fact that we're taking all of that grunge work off the table for them. If you're an IT department and a customer, you're looking at for how you can deliver more value to the business, right? And patching our software, upgrading our software, being responsible for hardware issues, that's not adding value to the business, right? Ensuring they're delivering the application SLA, ensuring the application is secure, helping reduce cost. Those are all adding tremendous value to the business, right? So the fact that we're able to deliver to them a cloud service allows them to sort of elevate the value that they can offer. And so that was one key insight. We wanted to bring all of those benefits to VMware Cloud on Dell EMC on-prem. The second thing I would say is just technically, it's a very different model to ship a customer's software and hardware and say you manage it, right? You're responsible for the SLA versus delivering a true cloud service, right? It requires a very different way to run your engineering team. It requires this thing called service ownership, right? That you're accountable for the SLA of your code running and production. You need to build out a site reliability engineering team. And it really requires very close engineering relationship between everyone who's working together to deliver that integrated cloud solution, right? And so we're taking all of those learnings and insights that we got sort of from our experience in the public cloud and now applying them with Dell to bring those same benefits to customers in the private cloud. One of the things that we've talked a lot on theCUBE about, in particular this week, is just how close the Dell VMware relationship seems. You have said, Stu, and you really know your stuff that this is the tightest you've ever seen it. And here you are talking about this jointly designed and engineered. Can you describe a little bit about sort of the culture of this partnership and how these two tech giants work together? Yeah, I can take a stab at least, I mean, yeah. Look, this is not new to us. We have been working together for a long time. But I think, as you saw in the keynote stage with Jeff and Pat together, this is a new level of a relationship. In terms of having our engineering teams work together, figuring out how to deliver the best customer experience. We already see that when we made an announcement three weeks ago with VxRail and VMware Cloud Foundations, being able to manage the entire life cycle right from the workload all the way down to the physical infrastructure using VMware Cloud Foundations. This is a natural extension of that model for us. We're taking some of the same engineering work, the tight integration, and then adding on another benefit of managing this for the customer and making things simple for them. And we think this is just the start. We think there's so much more goodness we can uncover for our customers as part of this journey. Yeah, I think it's great. The only thing I would add is the analogy I like to use is sort of like weightlifting, right? So this is a muscle that we've been building between VMware and Dell for many years now, right? And delivering a full cloud service on top of Dell hardware, that's like bench pressing 200 pounds, right? So if you just like had never worked out before and someone gave you 200 pounds of bench press, you probably wouldn't be successful. Now the good news is we've been working together for a number of years now. We've been building that muscle together between the two companies, right? VMware on VxRail, VMware Cloud on VxRail. And so now we're taking this next step forward. Hey, maybe we're going from benching 150 to benching 200. We have the ability to get there, right? And so in many ways our ability to be successful at this is based on the fact that we have been working together so well for a number of years now and building on that. Okay, so Veroon, we look at these different solutions in the marketplace in this space and sometimes it's a little tough to differentiate them because you look underneath the covers and you've got a lot of hardware geeks, you know? I'm one of them, I'm opened up the back of the cabinet and show me, I'm like, oh, I recognize that box and I do this, but like, say for example, if I go talk to Microsoft and I look at Azure Stack, they're like, don't really think about the Dell server underneath there and the partner thing I have. This is Azure. So when you think about the operating model, when you think about the consumption model, when you think about the applications, this is Azure. What I've had a little bit of trouble and trouble and I'm hoping you can help explain is, I think it's a similar type of story, but there is no Dell EMC public cloud. There's VMware in a couple of environments. So is that the right model to be thinking of? I mean, this is as a service, it's a consumption model, but are the applications similar to what I had if I've built a stack with Dell and VMware or give me the compare and contrast as to what I've done before and if some of the other options out there. Great question and I think it's something a lot of customers ask us as well. Look, I think this is a very unique offer compared to what we've seen in the market recently for a variety of reasons, but the first thing I'll start with saying is that customers today are already using VMware and Dell EMC for their existing workloads, right? This is essentially the same platform, right? So the tools that they use today, vSphere, HCX to migrate workloads, you know, NSX, vSAN, they are going to be able to carry forward all the work they've done there on this platform, right? So it's no different from that perspective. So the learnings they have, the processes, the automation, the ecosystem of backup disaster recovery that they use today, they're going to be able to use later as well with this as well. So this is less disruptive for them. So that's the first thing. The second thing I'd say is, you know, we think we have a unique advantage because we have a long heritage of working with customers in their data centers, whether it's VMware or Dell EMC or us combined together, being able to manage the complexity, the thousands of variables in a data center that a customer has where the things are not just homogenous, everything is not standardized. It's a very, very different problem from talking about a homogenous cloud data center where everything is standardized, everything is built for automation. We think we have a unique capability to be able to do that, and not only from a day zero or D one perspective, also from a support perspective.  We have a support organization across VMware and Dell EMC already built today. For scale, every single country, wherever people's data centers are. Again, a different support model, we think this will be a journey for folks who don't have that built out. And then finally, you know, I think I'm biased, but I think infrastructure matters. If you're going to take a bet on this platform for your edge locations, your retail locations, your thousands of retail locations, sure it's a fully managed service, but you need to have the peace of mind that this is going to continue to work for you. Even in a fully managed scenario, it is disruptive if there's hardware failures.  There's more than 4,000 customers that VMware and Dell EMC have jointly driven success with. So we think these are going to be unique factors that will create value for customers. Okay, so for the support model, I understand, the question we've been talking about, the various solutions in the portfolio is, their Nirvana is that cloud operating model that I don't need to worry about what version it's running, whether the latest security patch is in there, because that's been taking care of me. Are we close to that? Are we at that? How do we, how's that look? I mean, that's exactly the idea. That's exactly what we're going to deliver, right? And you know, that's powerful for the reasons you articulated, but even more than that, I would say, you know, it's an amazing vehicle for us to deliver value and innovation to our customers, right? You know, traditional model, hey, VMware develops software, it takes a year or two to develop, we deliver it to the customer, they take another six months to a year to upgrade it. You know, it's two to three years' latency between when an engineer has a good idea or a customer asks for something before they can reason we get to take advantage of it in production. With this new cloud delivery model that we're building together, that latency shrinks down to potentially just weeks, right? Because we are upgrading that service on a continuous basis. We can push those new innovations to our customers much more rapidly and they can immediately begin consuming them. Like literally those new features just show up in the service just like on your iPhone or you know, whatever other service you might be using. Same model can now apply to the data center. So it's an incredibly powerful thing for our joint customers. It's also really exciting for our joint engineering teams, right? You think about an engineer, they take pride in seeing the value of their work being used by customers and we can take that from two to three years to two to three weeks. That's a tremendous thing. Real instant gratification which makes for a happier employee, which makes for less turnover and all of that. Yeah, you got it, yeah. Great. Well Mark and Maroon, thank you so much for coming on the show. It was great having you. Great thanks for having us. Thanks for having us. I'm Rebecca Knight for Stu Miniman. We will have much more of the Cube's live coverage of Dell Technologies World coming up in just a little bit.