Good evening. Good afternoon and good morning wherever you happen to be in the world. My name is Robin Lord I work for a digital marketing agency called error here in the UK And I'm going to be talking to you about estimating opportunities with different searches Now it's really important that we're able to estimate opportunity because it gives us a way to talk about how valuable our projects are To other people and it helps us to figure out where we should be focusing our energies, right? So being able to know how much opportunity we might get in different places is pretty crucial So we're going to start with the most basic way of approaching that and then we're going to work our way up to a slightly More complex slightly more nuanced approach But to begin with the kind of things that we have when we start to estimate opportunity with different searches are usually the number of Keywords that we want to target for example Here's one setting up a business and the search for you that we have for those keywords So here in this case setting up a business. We think is searched at 6,500 times a month So it's 6,500 times a month so I'm searching for setting up a business now We're not going to assume that every single one of those is going to come straight to our website Right, we're not assuming that we could get 6,500 clicks straight to our website We get an idea that probably most people are going to click position one But some people are going to click position two and some people are going to click position three in position four and so on and so on So even if we were in the best possible position, we can't be everywhere So we can't just assume we're going to get that 6,500 clicks through to our website and the way that we start to handle that is by using something called a click-through rate Which if you haven't encountered it before looks roughly like this and you can probably see why it's called a click-through rate curve here We've got the curve We've got all of our positions one through ten here at the bottom and the side is How likely we think someone is to click through to our site So the easiest way to kind of visualize it is in a graph But what we what we usually have is in a table so that we can use something like a V lookup If we are saying position one then we know that our click-through rate for that is probably going to be about 35% So we use that to estimate what we could get out of this search for position one Likewise if we're looking at like position eight, we might say click-through rate for that is more like 5% So if we're in position eight, that's kind of what we assume we're going to be able to get out of it And we use that in a in a formula kind of like this one So we say the search volume so how many how many times something's being searched here at the top Multiplied by the click-through rate is how many clicks we expect to get so in this case 6,500 is our search volume 35% for position one gets us about 2.3 thousand clicks a month or 27,000 clicks a year you can see here You can kind of mark it here at this line just to visualize for you. That's kind of what we're doing here We're assuming position one will get us this kind of click-through rate and that would result in about 27,000 clicks a year for this search term Now we could stop that and some people would stop there in terms of okay well I'm going to assume I could get position one for every keyword and I'm just going to multiply out to make sure that I'm not making wild predictions about how much traffic I might be able to get but that's the way I'm going to leave it and that can give you a slightly more accurate estimate, but It's not factoring in times when we might already be ranking a bit for a keyword, right? So say for example, we are already ranking position eight For this specific search now we can see here position eight is already getting some clicks So if we if we move up to position one sure we're going to get more than we're currently getting But we're not going to get 27,000 clicks more than we're currently getting we're going to get a bit more than we're currently getting But we need to factor that in because there could be other searches for example that have smaller Search volume, but we're not ranking at all So the opportunity for us is bigger because really what we want to focus on is how much more Could we get then we're currently getting at the moment? So how could we handle that? Well fortunately the matter on that is actually fairly simple It's exactly the same sum as we've done here. We just need to look at position eight Figure out what our click-through rate is for position eight and then do that same sum to figure out how much we could get now in this case 5% gives us 325 clicks a month about 4,000 clicks a year So we're getting 4,000 clicks a year at the moment. We could get 27,000 clicks a year So we just subtract what we're currently getting and we say okay, we've got an opportunity here of about 23,000 clicks a year So that's starting to get us a bit more of a nuanced idea of what our opportunity is in different places because we're not Going to keep pouring energy into something we're currently doing kind of well for and we can start to focus on the areas where there's kind of untapped growth there How about we push things a little bit further still so we've got this Understanding of how much we could get if we got to position one say for this search Position one is a government website and position two is Amazon and position three is Google now Google once it has a government website once it decides a government website is quite relevant Doesn't really like to replace that because they tend to be very trustworthy. They tend to have pretty good information Also, it doesn't really like to replace Amazon by and large because Amazon tends to be a great result for a whole bunch of different things Particularly doesn't like to replace itself So if we're starting to think about what we could get for this keyword and we're assuming we could get position one It's 35% click-through rate. We might look at some of these results. I think actually I don't think I can get up here I think we should probably be looking a little bit further down. So maybe we look at something like position Say position four is a page about how to start a business position five is Wikipedia Position six is someone offering to offering to sell a service so that you can buy business setup and Seven is 50% off business setup. So they're again offering to sell, but they've got that cheeky 50% discount So if we look at all of this we might start to think okay, well, I don't think I can Reach position one probably not position two probably not position three and actually I think I probably Don't want to assume I can replace Wikipedia here either in position five So the best position that I'm looking at here is actually Position four. So I should go back to some of my click-through rate Estimates I should estimate based on position four rather than position one and then use that to get an idea of the total Opportunity we could get for this keyword now. You could be wondering how you might do this Actually, there's a bunch of tools that will give you the full top 20 Export for a whole list of keywords. The first time I did this I use the stat top 20 export I just exported everything and I dumped it into a Google sheet So I ended up with all of my keywords and all of the top 20 results row after row after row after row and Along my sheet Then I used a formula pretty simple formula to just find any Websites that we thought we wouldn't be able to beat and I use that to mark those rows any time We thought we would be able to be and then I just deleted those rows So I ended up with each of my keywords. I only had the ranking positions That we were at we thought we were actually able to achieve So in this case four six seven and eight which we are currently and then I just found the Highest position that we could get in that list for each keyword and I use that With my click-through rate curve to try and estimate how much traffic we might be able to get So this all really boiled down to like if formulas and V lookups So it's very accessible for anyone who wants to once again involved in this kind of thing now We could leave it there that would give us a pretty solid understanding of Where we might be able to get and it's far more nuanced than the picture that we might get at first blush If you want to get a bit more fancy with it There's other information that you could pull in to your analysis So say for example You're able to pull in the titles of all of the pages that are currently ranking say in position four It's how to start a business and position six. It's by a business setup in position seven It's 50% off business setup again if we look at these We could do some fairly simple formulas even just in Google Sheets to try to categorize these into different Intents so how to it's fairly clearly an informational search An informational result rather, so we're not trying to categorize the search We're trying to categorize the individual results By is a fairly clear purchase search and 50% off again suggests that it's trying to sell us something So if we start to look at these results, we can filter them down even more if we want to so say for example We're working on a site and we know it has to be a product page if we're going to target this term We have to use a product page. Well, we could do that same filtering process except this time We're marking anything where the title is something like how to or top tips or instructions how to or anything That seems like a blog post and we remove them as well So we say actually we couldn't get this either this is the highest we could get with this product page If we do that across all of our keywords We know that we have to use product pages and we're saying okay for this we're position six Well say position six actually has a pretty low search click-through rate If we're looking at position six for this keyword, we're already at position eight So maybe the opportunity that we've got for this keyword is actually pretty small And we decide that we want to focus on other things So that's another quick way to filter all of our different opportunities by just Removing the kind of results that we wouldn't want to compete with Alternatively if we don't already have an idea that it's got to be a product page We could go through all of our searches and say we know that it's going to be positioned for is the best place that we could get We can use that same category. We can use that same categorization formula to say well position four looks like it's informational So as we're as we're identifying our opportunities We can quite quickly say when we go to do a content brief for this for example This needs to be a blog post. It shouldn't be an update to a product page Now the aim for any of this isn't to make people's decisions for them We're never going to just send this sheet for example after someone and not have an SEO professional Look at it, but it means that instead of spending lots of time having to reverse engineer having to think Really hard about all these different things and pull information into one place We have a starting point for people to go from so when we have someone who's an expert SEO looking at this They've got all of the information in front of them to begin with that's actually the approach that we tend to take in general in Error so when we do this kind of work We actually tend to use a Python script and that script pulls in all of the top 20 results It also categorizes those search results the individual results rather than the rather than the search term itself based on whether they seem information or transactional and it finds that opportunity it finds that highest place that we could currently get and Subtracts where we're ranking at the moment and that means that we end up with a sheet where we can order things quite nicely Based on highest opportunity to lowest opportunity and based on and categorize them based on what kind of results They are we also pull in some other things like authority Which is one thing that you could also use to filter down your results if you're starting to dig into this And we also use the search results to see how similar different searches are as a way of clustering them So those are some other things that you can dig into once you get familiar with these kind of concepts and really start to Accelerate now you don't have to go that far You could do any step along this route and get a little closer to a nuanced understanding of what this search result could get you And that's going to be a really positive advancement because the more that we can bring in this nuance The more quickly we're able to identify these different things all of these are decisions that you're going to be making anyway You're already a smart SEO professional. You're already going to know all this information It's about speeding up your path to that answer. Anyway, thanks very much for listening to me I really enjoyed chatting this through with you. I hope you enjoyed it as well, and I'll look forward to hearing your thoughts Thanks a lot


Welcome back to Investor Intel at PDAC 2018. I'm pleased to be joined by Mark Chalmers. Mark is the CEO of Energy Fuels. Mark, Uranium and Vanadium have both been in the news a lot lately. First of all, can you comment on what's going on in the market and particularly what's going on politically in the U.S.? Well look, Uranium itself has historically been a political kind of element. I mean, we as a company with U.R. Energy, we filed a Section 232 trade petition to limit imports of Uranium into the United States. Now, Section 232 is quite in the media a lot right now with the steel and aluminum, so I think people didn't know what Section 232 even meant up until a few days ago when the Trump administration, after a review by the Department of Commerce, is looking at imposing tariffs on imports from various forms from different countries. So we as a company, we actually started this process about a year ago and we felt that the low-level production in the United States, the United States is the largest consumer of Uranium in the world, but we're only actually producing less than 5% of our requirements. So the Section 232 is a trade act that is driven for any type of commodity like Uranium or Vanadium or in the cases of steel or aluminum that may have national security implications. And so that is what we've put in with U.R. Energy. Well, I think that would be a pretty compelling case that Uranium is an important strategic element. On that basis, you guys are already a producer. Obviously, a U.S. producer would be extremely benefited by any kind of ruling under 232. Yeah, we would be benefited naturally, but the thing is that we want to be reasonable with this Section 232. We want to survive as an industry. The nuclear industry in the United States has been challenged with natural gas and the increase in renewables lately. So we don't want to be greedy here, but we do think that a front end of the nuclear fuel cycle is required in the United States. So it's really, again, when we talk about national security issues, it's not just Department of Defense, it's really more focused on the nuclear power generation industry for the electricity. Because 20% of our electricity comes from nuclear. Well, sometimes people don't regard Uranium production as being something that's particularly green. But from my understanding of your business, you guys provide some very green services not only in how you process your own materials, but also in cleaning up other projects or other mining operations. Is that correct? Yeah, we have our white mason mill in Utah. It's the only remaining conventional mill in the United States that's operable. And we have two sources of revenue with that facility. We have what we call alternate feed, which is basically taking out of spec material or low level material that we can actually run through the mill and recover the Uranium and repackage, we're basically recycling products that had Uranium in it. In a lot of cases, returning that to wherever we got it from, like it could be a major other mining company that had a stream that was unusable otherwise. And that's what we call alternate feed. But in addition to that, we are also as a company looking at doing cleanups of former Uranium mines, most of which were mined in the 50s and 60s during the Cold War, largely subsidized by the United States government. And the EPA has collected around two billion dollars of cleanup funds in trust on the Navajo Nation alone. So we're looking at using our facility which is just a short distance from the Navajo Nation and providing a service where we can take low level radioactive material left from that period of time, the 50s and 60s, recover the Uranium from that material and dispose of it safely in our facility. Well, there's a lot to the story. What should investors be looking for right now as a reason to own energy fuels? Well, I mean, we're primarily a Uranium-focused production company. That's number one, the Section 232 and what that could do to our company in terms of stimulating and providing an inflection point. If that review, that petition, which we think will happen will go forward. And also the Vanadium space because we have historically produced quite a lot of Vanadium from the assets that we have. Over the course of the last 38 years, we've produced over 45 million pounds of Vanadium at current prices that's in excess of $500 million of Vanadium. So there's some real interest in Vanadium for steel hardening which has been the normal use, but also Vanadium flow batteries for renewable energy storage systems. Well, Mark, thank you so much for joining us today and we'll look forward to seeing you again soon. Thank you very much.


If you look closely at the healing ministry of Jesus, you'll notice something that he does that helped the people to be led to their moment of healing. And I want to talk about the moment of faith. You see, there comes a moment when their faith is at its fullest and you need to learn to seize that moment so that you can be equipped to minister God's healing power. I'm going to be teaching on this edition of Spirit Church about the moment of faith, and this is a continuation of my series, How to Heal the Sick. But before we get into this lesson, Stephen Moctezuma is here with me. He's going to lead you in some anointed worship and then we're going to get right into this lesson here. It's Stephen Moctezuma.  So I'm preaching on God's power to heal the sick and I'm actually giving keys to the healing ministry much like I'm doing in this series. And as I'm ministering to the people, I just see this woman in her wheelchair and she's seated about the third row and I could not continue my sermon because I felt the Holy Spirit stirring me to talk to this woman. So in the middle of my message, I stopped teaching on what I was teaching on at that moment and I began to speak life into that woman's situation. I remember telling her something to the effect of, you know, a lot of people would say, I'm cruel for telling you that you could be healed.  But I told her in boldness because the Holy Spirit had led me to do this. I told her that you're going to walk out today without that wheelchair. Now it was a moment that was inspired by the Holy Spirit but I saw faith in her eyes. There was something about the way she was listening intently to the word that I knew deep within my spirit that this woman was ready to receive her miracle. And so I made that declaration in faith, finished up the message and prayed for her after the service. By the end of that service she was in fact walking and in fact just maybe a couple weeks later she was dancing at a wedding. She sent us a video from her home walking around her house and God had restored her. Even some injury that had been on her body was visibly shrunk and the doctors were amazed at how rapidly she was healing. So the healing power of God came in that moment. There was another man or another individual in a wheelchair that I ministered to in the Midwest. This man was paralyzed on his left side completely. He had been paralyzed for several years and he was in a wheelchair. He could not walk, he could not move. He had trouble feeling in specific parts of his body on his left side. And I remember the service had just begun and usually the way I start a service is the worship will go, I'll come out, lead a prayer to open up the service, then let people go back to their seats. But in that break between the worship and the start of the message, as the people were continuing to their seats, the Lord pointed out that family to me. And the wife who was wheeling her husband up to the front looked at me and said, can you pray for him? She wasn't close enough to the stage to where I could hear her, but I could read her lips. She's asking, can you pray for him? And I remember I was going to send them back to their seat and pray for them after the service because there's a certain order to how I do ministry and I have to do it orderly because that's I believe how God blesses it. But I also have to be obedient to the voice of the Holy Spirit spontaneously. So this woman brings her husband up and she told me, will you pray for him?  I said, don't you let that woman get back to her seat. You pray for that man right now. And so in faith, I told them, that man is coming out of that wheelchair today and he's going to walk and we're all going to see it. The crowd was stirred with faith and I was thinking to myself, I sure hope I heard God on this one. And so she brings up this man, paralyzed, couldn't walk, couldn't move hardly, hardly any feeling. And we began to pray for this man and as we pray for his legs, I was praying for his legs first. I'm praying for his legs and he began to kick his leg up out of that wheelchair. And with each time he kicked his leg, the people were amazed and began to praise the Lord. And so eventually this man is walking back and forth across that stage for the first time in years. His wife is in tears and Jesus received the glory for that healing. But there was a moment of faith. Another instance, I was on Facebook and believe it or not, somebody commented, please pray for me. I need a healing. I believe that you can pray for me through Facebook and I'll be healed. And again, I saw faith in that moment. I didn't even pray for them verbally. I wrote back a comment, receive your healing in Jesus' name. That person within minutes commented back that they had been healed of their ailment because they stepped out in faith. It was a moment of faith. You see in the healing ministry as you minister God's healing power, you're going to see opportunities and moments that you have to seize. There are rivers of God's healing power that flow. There are streams that come but for a moment. And you have to be sensitive enough to the Spirit to recognize and read the moment and recognize and read the faith in the people. You see in Matthew chapter 9, verse number 22, the scripture tells us very plainly what it was that caused people to be healed. Jesus turned around and when he saw her, he said, daughter, be encouraged. Your faith has made you well. Again, in the book of Luke, this time in chapter number 18, verse number 42, we see something else very similar. And Jesus said, all right, receive your sight. Your faith has healed you. Romans chapter 10, verse 17 tells us faith cometh by hearing, hearing by the word of God. Now, in context, that scripture in Romans is talking about the hearing of the gospel message, stirring faith to receive salvation. But the principle can also apply to healing in that when you hear a truth, it inspires faith within the soul to receive that truth and manifest it in the material world. You see, if you can believe for it in your spirit, it can be manifested in your reality. But there has to be the moment of faith where you seize it. Now, in Mark chapter 6, verses 5 through 6, we see that Jesus could do no miracles. Why? Because the people were unbelieving. They had no faith. In Luke chapter 8, verse 43 through 44, we find that a touch healed a woman. In Luke chapter 7, verses 2 through 10, we find that the Lord sent a word to the Centurion servant. Think about the blind man who was healed when Jesus took mud, raised it to his mouth, spit in it, and put it in the man's eyes. Now, I don't know about spitting in mud unless you're really led of the Holy Spirit. Don't do it. But Jesus, led perfectly by the Spirit, did something that to us seemed rather odd. I believe it was a reference to his power over creation, but just as God breathed life into the dust of the earth, so I believe Jesus spit life into that mud and healed this man. So I think it was a parallel to creation. Whatever it was that Jesus was doing or trying to communicate, the point is that it was a touch for a woman, it was the word for the servant, and it was mud for a blind man. Either way, what Jesus did is he worked with the faith that the individual had. Now, I've been in services where the minister will get up and they'll just right away start trying to pray for the sick. But if the people's faith is not present, there is no sickness that is going to be healed. It's just the truth. Now, I understand that there are those who go out into the streets and minister and I've done it myself and they lay hands on the sick and that's powerful when that is done. But even those moments are led by the Holy Spirit and organized by God. My point is simply that you have to be able to discern faith in someone and capture that faith and minister in a way that they're going to be able to receive it. So I'll give you another example. I was up, I think it was in the Redlands in near Los Angeles. And so I drove out to the Redlands. The place was packed. They had bused people in from all over the place. The place was so packed that people couldn't even get inside. They had to set up chairs all outside of the parking lot. They almost filled up the parking lot with seats. It was a powerful, powerful draw that had been on that service. But I went in there and I remember I ministered and taught and taught and taught and taught. And at first where I felt faith, the more I taught, the more I felt the faith of the people dwindling. And I thought this was odd because I said, Lord, the scripture says that faith comes by hearing and hearing by the word. Why is it then that the more I teach, the more I talk, the less I feel faith in the room. The more the atmosphere changes, one of doubt. And I couldn't figure this out until later. I realized that I had missed the opportunity. I had failed to seize the moment of faith and I had talked past their ability to receive. Now I'm not saying that in a condescending way, but the truth is that people reach a moment where their faith is full and you have to reach out and grab right then and there. You have to in that moment, stir that faith and then seize it as soon as the opportunity presents itself. And this is only by the Holy Spirit. People ask me, how do you know when you're being led by the Holy Spirit? How do you know the voice of God? Well, I asked the question right back as an analogy, which is, how would you explain sight to someone who was born blind? How would you explain sound to someone who was born deaf? You see, your senses, though you experience them, cannot be described to someone who themselves does not experience that sense. In the same way, I cannot explain to you. I cannot describe adequately with words how the voice of God sounds or what the voice of God sounds like in any given situation. All I can tell you is that it's a different sense.  And so the way you find this sense is by number one, reading your word daily. Number two, being in prayer constantly. And number three, being aware of the presence of God in every moment of every day. And when you walk in those three things, what you'll notice begin to happen is your spirit sense grows stronger. Your spirit sense becomes more keen. And you're able to sense things like those moments to seize those opportunities and help people receive their healing. So don't minister beyond their moment of faith. Don't miss the fullness of faith. Instead, work with them where they are. Don't teach theology when you're trying to minister healing to a new convert. Give them an analogy. Give them a story. Jesus often told parables and stories. Why? Because it helped to relate to the person he was ministering to. He did not go for the most part to the crowds quoting chapters and chapters and chapters of scripture, especially to Gentiles. What he did is he told parables. He told stories. He used agriculture and he used different structures of society as examples in which he would place his stories that proved his points or demonstrated his truth. And in the same way, we must work to relate to the ones to whom we are ministering healing. And so look for the moment of faith when you're ministering God's healing power. Watch the person. Don't just go in there and have a template or a system of how you want to minister to the sick. Don't go in there with the same point, one, two and three. Go in there being ready to be led by the Holy Spirit, equipped with God's word, discerning of the moment and sense where God is leading you to minister to that one individual. And as you seize the moment of faith, you're going to notice that miracle after miracle follows you wherever you go. You're going to notice, you know, and I'm not saying this, please don't hear what I'm not saying. I'm not bragging, but you don't know how many people tell me, I've had so many pastors lay hands on me to receive my healing and it never happened for me.  This key right here is something that most believers don't know about. It's finding the moment of faith. And this I believe is one of the ways you're going to increase the potency of the healing power upon your life. That is it for the lesson. I want to pray with you now. I want to pray that God would help you to discern the moment of faith. And then I want to pray with you for your healing. Come on, let's believe. Stretch your hands toward mine in faith and believe that God will touch you now. So Father in Jesus' name, I pray for that one receiving this prayer now. And I ask you Father in the name of Jesus that you would give us eyes and ears that are discerning. Help us Lord to recognize the moment of faith that we may seize it and manifest the miraculous in the precious name of Jesus. Give us discernment Lord. Father, I pray for those believing for a miracle in their body. I pray that healing river begin to flow now in the name of Jesus. There's somebody watching me. You're hearing me. There's an issue with your lungs. You have trouble with your breathing. And I just, you're sensing like a warm heat go through you right now. The timing that you're watching this video is perfect. It's the Lord's moment to heal you. And even right now I can feel this that issue with your lungs you're going to find that it's changing and there's a miracle manifesting your body. Go back to your doctor. Have him check you. I believe the Lord's healing you. Father, I thank you for the miracles that are happening all over. Lord, I pray for deaf ears to be open in Jesus name. I pray for blind eyes to be open. I can sense a tremendous anointing right now. Lord, I thank you for the healing power of God flowing. And I give you the glory Jesus for every miracle. Every miracle. Lord, your healing touch destroys the power of sickness over our lives. We bind sickness now in the name of Jesus. I want you to say it because you agree. Say amen. Well, I want to welcome now the new members of Spirit Church. There you are up on the screen. We love you. We're praying for you. I always say that because I always mean it. If you like information on how you can join the Spirit Family, simply visit the website at the bottom of your screen. Now Spirit Church membership is 100% free. What you're going to get every single week is a brand new, fresh, anointed teaching in your email inbox every Sunday morning. And you can reply to that email for prayer support from our ministry staff. On top of that, you're going to be joining the Spirit Family, which consists of over 3,000 members worldwide. I want to read your comments now. Your comments here I'm reading are from last week's teaching, which was How to Heal the Sick Part 3, Defeating Doubt. Here's what some of you had to say about that message. The first commenter writes, I always feel blessed with your teachings and preaching. Thank you so much, Brother David. I thank God for making your ministry known to me. I'm praying that one day I could attend your church.  By the way, we have some information on that. Not starting a church, but for those of you who don't know, we're in the middle of a campaign to build a brand new TV studio and that's where you'll be able to attend in person. There's some details on that coming up at the end of the broadcast here. The next commenter writes, I'm blessed by listening to you once again. Yes, I was facing doubt. God is so timely to speak to me through you. You are so anointed man of God, blessings all the way from Pakistan. Joseph writes, Thank you, Ome God bless you, Brother David. Your teachings are a blessing each week. They help me in my life and they help me to take my next great step in my faith. Scott Warren writes, Praise God for you, Brother David. This message really touched me. Also, Stephen, praise God for you, Brother. Amazing song. I love it. It's been a song that's constantly on my heart. For those of you who haven't done so yet, I recommend you check out Stephen Mock. This was a worship playlist here on EncounterTV. And the final commenter writes, Praise be to God. Thank you for another teaching that has stirred my faith. I always feel the presence of God and the power of the Holy Spirit in every teaching that you share. I am claiming to be free from doubt. May the Almighty God continue to empower your life and your ministry to win more souls. I can't wait for next week's lesson. God bless. Well, I know that that is what makes this channel unique is that people can fill the presence and power of the Holy Spirit. Really, this is the Holy Spirit's channel. If you look behind me, you'll see that logo of our ministry. The logo consists of two symbols, the fire and the dove. Both symbolize the person of the Holy Spirit. Now, I want to talk to you about what we're doing as a ministry to expand. Now, don't turn me off yet. Some of you, you've never gone this far in the video, but I want you to stick around. You're watching this for a reason. It's not by accident. We are right now in the middle of a campaign to raise 1,000 new $30 a month supporters. That would bring in $30,000 extra monthly for the ministry. That was the campaign. 1,000 was our goal. Look at how far we are now. Let's finish this up. Let's get this done so that we can start looking and for places to lease. We actually already are looking for a place to lease. So get signed up. That's where we are in our campaign. And that monthly cost is going to help with the lease. It's going to help with the insurance. It's going to help with the maintenance. There's a lot of monthly expenses that go into a building. So sign up to become a $30 a month supporter. What we're going to do with that facility is really simple. We're going to do more broadcasts more often in higher quality. We're going to be a live broadcast from the studio on a regular basis. We're going to build a place where a studio audience can come in. This means that you can come in and listen to the tapings live. Steven Moctezuma will do worship. I'll teach a lesson and we'll actually be able to lay hands on you when we pray at the end. So I know it's going to be a powerful tool. It's basically a world evangelism center. And I need your help with building this. It's also going to enable us to do more events more often in more places. So quite simply, what is our why? We want to win more souls. How do we do that? Our ministry wins more souls through events and media. And this new ministry facility is going to help us do more events and more media in higher quality than ever before. So if you sign up to become a $30 a month supporter today, I'll send you a copy of either Carriers of the Glory or 25 Trues about demons and spiritual warfare. That will be our initiation gift to say thank you. Sign up today if you're watching this on the YouTube app. Just wait until the very end of this video. There'll be a red button that you can click. It will take you right to where you can donate. 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If z is a function of two variables, then the directional derivative along a unit vector a v is going to be given by and corresponds to the rate of change of our function in the direction of v. We can express this conveniently as the dot product of the vector whose components are the partials with respect to x and y and the vector corresponding to our unit vector. What if we have a function of three variables, g of x, y, and z? Through a similar analysis, we obtain the following. If v is a unit vector, the rate of change of a function in the direction of v is our directional derivative and it's the dot product of the vector whose components are the partials with respect to x, y, and z with our unit vector. And in fact, this generalizes two functions of four, five, six, or more variables. Additionally, the direction of greatest increase of our function will go in the direction of this vector consisting of those partial derivatives. So for example, if we have a function of three variables, we can ask how rapidly g is increasing in a particular direction. Again, since our directional derivative is like the slope and it depends on how far we go, we want to find a unit vector. So we need a unit vector in the same direction as three, four, twelve, so we find the magnitude. And we'll scale by one-thirteenth to get a unit vector. So the directional derivative will be the dot product of the vector whose components are the partial derivatives with the unit vector. We know what the unit vector is. We just need to find those partial derivatives. So we'll find the partial of g with respect to x, with respect to y, and with respect to z. Substituting those in, we find the directional derivative is, and our directional derivative does depend on where we are, so we actually need to know the coordinates. So we know that x is equal to two, y is equal to one, c is equal to negative four, so we can substitute those in and find our directional derivative. For example, suppose we wanted to find the direction of greatest increase. So remember that the direction of greatest increase will be the vector consisting of the partial derivatives. So let's say if my function w equals x cubed minus sine xy plus z squared, so we can find the partial derivatives. Now the actual value of this directional derivative depends on x, y, and z. Fortunately we know what those are. If x equals five, y equals zero, z equals three, then these partial derivatives have values. And so w will be increasing most rapidly in the direction of the vector 75, negative five, negative six. Let's think about this a little more. If going in the direction of the unit vector a, b, c gives the greatest possible value of dvf, then we know that the directional derivative will be greater than any other directional derivative for any other unit vector a prime, b prime, c prime. But if I multiply everything through by negative one, this changes the direction of our inequality, which means that if we go in the opposite direction, our directional derivative will be less than the directional derivative of any other direction. And so if a, b, c corresponds to the direction of greatest increase of our function, then negative a, negative b, negative c corresponds to the direction of greatest decrease. And this may be important. For example, suppose a temperature at any point in a room is given by some function, a particle is at some point five, three, one. And let's find its temperature and in what direction should the particle move to decrease the temperature most rapidly? So our function gives the temperature at any point x, y, z, and we know the x, y, z coordinates, so we can compute the temperature. 440 degrees, we're essentially talking about a room on fire. Now if I want to find the direction of greatest increase, I want to find the directional derivative, so we'll find those partial derivatives. So we'll find our partial with respect to x at the point five, three, one. Our partial with respect to y at five, three, one. Our partial with respect to c. And so this gives us the direction of greatest increase. But since we actually want to look for the direction of greatest decrease, we should move in the opposite direction.


In the last 8 lectures, we covered 2 major topics that is the seismology and seismic inputs. In the seismic inputs, we have seen that we have different forms of the seismic inputs and the one that is used depends upon the analysis at hand. Out of the different forms of the seismic inputs, 3 different seismic inputs are quite often used. First one is the time history record of the ground motion, second one is the response spectrum or rather the design response spectrum and a third one is the power spectral density function of the ground motion. For designing the structures for earthquake forces, generally we analyze the structure using the response spectrum method of analysis in which the response spectrum becomes the input. When we model the ground motion as a random process that is the future earthquake considered as a probabilistic process, then the power spectral density function becomes the input and we analyze the structure using the principles of random vibration. The time history of ground motion is used for the deterministic analysis of structures for earthquake and in that we are interested in the response of the structure at every instant of time t. The time history ground motion can be Fourier synthesized and the Fourier contents of the ground motion can be obtained as we have seen before and therefore, the two different types of analysis that can be performed for a specified time history of ground motion. The first one is a time domain analysis in which we obtain the response of the structure with the time history of ground motion as input in time domain using certain time integration scheme. The second one in which we use the frequency contents of ground motion and perform the response analysis in frequency domain and then we perform an inverse Fourier transform to get the time history of response of the structure. Now, both these topics that is the time history analysis of ground motion in time domain and frequency domain will be discussed in the subsequent lectures, but before I do that let us try to look at some of the fundamentals of dynamics. I am sure you are quite conversant with them, but still for recapitulation let me take up those. First the equation of motion for a single degree freedom system is known to all of you. This is a mass of a single degree freedom system. It is connected by a spring to a support and a dashpot also is connected to the support. Dashpot indicates the damping and the spring denotes the restoring action of the elastic system and the M denotes the mass of the elastic system. Now, if the mass moves in this direction x then an inertia force is developed due to the oscillatory motion of the mass. All of us know that for a oscillatory motion the acceleration always is in the opposite direction to the displacement direction. Therefore, if the displacement takes place in this direction then the inertia force which will be developed will be equal to mass times acceleration acting opposite to the direction of the motion. Similarly, if the mass is displacing in this direction then the restoring force developed in the spring will be K x and this will be acting opposite to the direction of the motion. The C x dot which is the damping force this damping force is called the viscous damping force is written as a coefficient is equal to a coefficient of viscosity multiplied by the velocity. This damping force by definition acts opposite to the motion because it dampens the motion. So, all the three quantities that is the restoring force the inertia force and the damping force are acting opposite to the direction of the motion and in the direction of motion the dynamic force is acting. Therefore, one can write down this simple equation that is M x double dot plus C x dot plus K x is equal to P t. These follows from the D L Ambers principle which is well known to all of you. Now, when there is a support movement then the equation is slightly modified. Let us see how it does get modified. Let us consider these portal frame the beam here is a rigid beam and at the center of the beam the mass of the entire system is lumped. There is a sway movement of the frame for a unit sway the force which is required is equal to 12 v i by L cube for this column 12 v i by L cube for this column or in other words 24 e i by L cube will be required to produce a unit displacement over here. So, we can say that the stiffness constant for this particular frame is equal to 24 e i by L cube. Now, this frame can now be modeled as a single degree of freedom system in this particular fashion the M mass is this rigid mass it is attached to a spring the spring has a stiffness coefficient of 24 e i by L cube it is attached to a dash pot in order to represent the damping. The since the base is accelerated in this particular direction the support over here is also accelerating in this direction and it is in the same direction as that of the motion that is what is shown over here also the acceleration and the displacement direction are in the same direction. So, this model is taken for that we write down the equation of motion. So, for writing down the equation of motion we make a slight change the change is that the mass now undergoes a total acceleration total displacement and total velocity. The definition of the total displacement total velocity and total acceleration they obvious from this equation we can write down the total acceleration or the absolute acceleration to be is equal to x double dot plus x double dot g where x double dot is the relative acceleration between the support and the mass the x double dot g represents the acceleration of the support. So, if we add them together we get the total acceleration similarly one can get a total displacement total velocity of the system. Now, the most important thing is here that the inertia force is equal to mass times the total or absolute acceleration if it is present in the previous case we did not have any absolute acceleration or the total acceleration because x dot itself was the total acceleration because there was no movement of the support. But if there is a movement of the support then the inertia force which is generated is equal to mass times the total acceleration of the system. So, that is what we write over here m x double dot a becomes the inertia force c x dot plus k x they remain as before because the restoring force in the spring will depend upon the relative motion that is the relative motion between the support and the mass and it is assumed that the damping force is also equal to the coefficient of viscosity multiplied by the relative velocity. Since there is no force acting over here it is set to 0. Now, if x double dot a the expression of this is substituted over here then keeping m x double dot term on this side we have a equation in on the left hand side of the equation all in terms of the relative quantities that it m x double dot plus c x dot plus k x and on the right hand side we can take mass times acceleration that is it because minus m x double dot g. Now, that we can write down as a effective force P t. So, this effective force P t that is minus m x double dot g that is called the earthquake force and this earthquake force is a mathematical construct it cannot be physically seen, but the effect of this force can be felt or are visible. Therefore, the earthquake force that we talk of is in fact a mathematical construct. Now, with this what you call the definitions or derivations let us look at this equation that is m x double dot plus c x dot plus k x equal to minus m x double dot g that is what we have proved or derived. Now, this can be written in three different forms first these entire equation can be written in terms of not absolute displacement velocity and acceleration, but the total displacement velocity and acceleration for that what we do is that in this equation if we add on c x dot g that is c x dot g if we add on over here and if we add on over here k x g then they turn out to be k x t because x plus x g is equal to the total displacement. Similarly, here it turns out to be x dot t and since we have added k x g and c x dot g on the left hand side we put them also on the right hand side thus the equation of motion now becomes in the form of total displacement. If we solve this equation we will get the total displacement of the mass rather than the relative displacement. One can note that on the right hand side now the requirement is the ground displacement ground velocity not the ground acceleration. Now, these two equations again can be written in the form of a state space. Now, the concept of state space is that a dynamic system at an instant of time t can be represented by displacement and velocity. If we know the displacement and velocity of a dynamic system at an instant of time t then we know its state in the time space because knowing the displacement velocity at an instant of time t one can also find out easily the acceleration. Therefore, we take x and x dot as the quantity of interest together represented by a vector called a capital X vector this denotes the state of the system. If we try to write down the equation of motion in the form of this new vector or the new state variable x then the equation takes this form x dot is equal to A x plus f. If we look at this equation we see that this is a first order differential equation rather than a second order differential equation. In fact, from the theory of differential equations one knows that if there is a nth order differential equation then we can write down that nth order differential equation into a set of n first order coupled differential equation. So, since it is a second order differential equation it can be written as a set of two coupled first order differential equation and that is what is seen over here. Let us now try to derive this A matrix and the f matrix. For that what we do is that we divide this original equation by m we have c by m k by m instead of c and k m on this side vanishes. Then you write down a equation which is a redundant equation x dot is equal to x dot then write down this equation in the form that x double dot is equal to minus k by m x that is we take both these two terms on to the right hand side and we get a equation like this. These two equations we write together by defining x dot x and x dot as a vector. Now these two equations can be written in this form x dot and x double dot as a vector is equal to this matrix multiplied by x into x dot where x and x dot is the state of the system or the value of capital X vector. Now if we expand this matrix then x get multiplied by 0 x dot multiplied by x dot on the left hand side we have got x dot. So, it is the first equation. If we expand if multiply this with this then we get minus k m into x that is this term and then we get minus c m into x dot this term. We add on to this 0 this will not be x double dot g this will be minus x double dot g. So, 0 minus x double dot g because the force there is no term over here forcing term over here that is why it is 0. So, we get the matrix a this vector f which is shown here in this equation. In the same fashion this second order differential equation expressed in terms of the total displacement can also be written in the state space where the a matrix remains the same as before there is no change whereas, the f bar that becomes x g x dot g because on the right hand side of this equation we can see that the force depends upon x g and x dot g. So, this f bar becomes this and this is multiplied by this matrix in order to get this total force quantity. So, the equation of motion can be written in the four different forms one is in the relative displacement second order differential equation then in terms of the total displacement again a second order differential equation then two state space forms one is in the form of relative displacement other is in the form of the total displacement. So, the solution of these four types of equations provide us the response of the system we can get the relative displacement as the response we can get the total displacement also as the response. Now when this is for the single degree of freedom system when we extend this to a multi degree of freedom system then what we can do is that in of the single degree of freedom system is replaced by a mass matrix instead of a single m it becomes a matrix x double dot becomes a vector of a n number of accelerations. Similarly, instead of small c we have a c matrix x dot again is a vector k is not a stiffness constant, but a stiffness matrix of the system x is the displacement vector here instead of writing down m x double dot g we write down matrix m multiplied by influence coefficient vector i into x double dot g. So, these are the modifications that is made when we try to write down the equation of motion for a multi degree of freedom system. The proof of this is well known to all of you those who have done the dynamics therefore, I am not going to you know derive these equations, but a simple way to write down the equation of motion for a multi degree of freedom system is to consider on an extension of the single degree of freedom system and in that extension we simply replace m c k by some matrices called mass matrix damping matrix and stiffness matrix mind you these stiffness matrix corresponds to the dynamic degrees of freedom and all of you know that the dynamic degrees of freedom differ from the kinematic degrees of freedom there may exist a number of kinematic degrees of freedom in a particular system, but all kinematic degrees of freedom are not the dynamic degrees of freedom by definition the dynamic degrees of freedom should have a mass attached to it. So, this k in fact is a condensed stiffness matrix of the structure corresponding to the dynamic degrees of freedom. So, what is generally done that we write down the full stiffness matrix of a particular system considering all the possible independent kinematic degrees of freedom and then condense it to the dynamic degrees of freedom for which we are able to attach some masses. So, this condensation is a must for most of the dynamic problem. Now, this i influence coefficient vector can take up different form for that i influence basically we can look at different kinds of problems, but before that let me also tell you that when you are solving a multi degree of freedom system subjected to ground motion then we categorize them into two distinct categories. One is the single support excitation and the second is a multi support excitation. What we mean by single support excitation and multi support excitation that we will see shortly, but if we have a two component ground motion then the i this influence coefficient vector this influence coefficient vector can be written like this. For example, this is a system in which we have got or let us say this is a system in which you have got two ground motion one in this direction other is in this direction. Then one can write down represent the acceleration by this formulation where i is a matrix of 1 0 0 1 1 0 0 1 so on. This is a vector of x double dot g 1 and x double dot g 2. If we multiply this vector by the first row over here then it becomes simply x double dot g 1 this becomes x double dot g 2 again this becomes x double dot g 1 then it becomes x double dot g 2 and so on. In a multi degree freedom system like this say this is a multi degree freedom floor of a multi degree freedom system then it has a acceleration in this direction and an acceleration in this direction. So, mass times the acceleration in this direction is one inertia force and mass times the acceleration in this direction is another inertia force. Therefore, for these two degrees of freedom the forces which are inertia forces which are acting is m x double dot g 1 and m x double dot g 2 that is what we get. So, this is for one floor this is for the next floor and so on. So, the i that is the influence coefficient vector that is what is shown over here influence coefficient vector that is i t is given by 1 0 1 0 and then 0 1 0 1 so on. And where x double dot g is represented by x double dot g 1 and x double dot g 2 that is what we have shown you. So, for the two component ground motion we get the i the influence coefficient vector in this particular form. Now, in a concept of single excitation single support excitation and multi support excitation is clear from these two diagrams. If all the three supports over here of a particular frame move by the same ground acceleration then we call this as a single support excitation and if the supports move differently that means the ground acceleration at this support is different from the ground acceleration at this support then we have got a multi support excitation. In fact, a multi storey frame if we look at it then the multi storey frame is subjected to a multi support excitation rather than a single support excitation. Because as the earthquake wave travels then the excitation that it will cause to the first column and the excitation that it will cause to the last column they will be different. In fact, they will be different because there will be a time lag between the two. Whenever there is some excitation at the first column at that time the ground motion has not reached the last column therefore, at that time the excitation there is 0. Therefore, we see that there is a time lag between these two excitations. However, the earthquake motion or the ground motion travels with such a velocity that in order to cross say a distance of about 30 meter or 40 meter or so the time taken is very small and therefore, the difference between the last column and the first column so far as excitations are concerned generally is ignored and we say that all the bottoms of the columns are excited by the same time history of excitation. So, this is an idealized condition and for most of the practical problems we consider this idealization and we say that the entire building frame is a single support excitation system. However, if there is a long bridge supported on the piers and the piers are at a distance of about 100 meter or more than that then in that case the excitations at the different piers supports are different and we consider the problem then as a multi support excitation problem. Similarly, for a very long structure like dams we have a multi support excitation condition therefore, we treat the problem as a multi support excitation problem. For three component ground motion in a similar fashion we can show that the influence coefficient vector will be of this type 100 then 100 then 010, 010 and so on. So, in this way one can write down the for the three component ground motion the influence coefficient vector but, we see an interesting thing is that that influence coefficient vector i whether it is for a single component earthquake or a two component earthquake or a three component earthquake they consist of generally one and zeros. Now, let us try to look at some of the example problems to illustrate how the influence coefficient vector is obtained for a single point excitation system. So, here we show two problems in this problem we have a ground acceleration in this direction we have a mass here and accordingly we have a degree of freedom in this direction the second mass is over here and the degree of freedom is this and the third mass we have got two degrees of freedom one is in this direction other in this direction. So, in this case the i the influence coefficient vector takes for u 1 it becomes 1 for v 1 it becomes 0 and for u 2 and u 3 they becomes 1. So, since there is no ground motion acting in this direction therefore, we have this form of the influence coefficient vector and the 0 coming into picture whereas, for this particular problem the influence coefficient vector corresponding to these three degrees of freedom are all once because the entire base is moving by the same excitation and therefore, all the degrees of freedom moves by the same amount and therefore, we have the influence coefficient vector as 1 1 1. Similarly, if we wish to write down the influence coefficient vector for these three dimensional frame then for single component earthquake obviously, we have the if the earthquake is having in this direction then we have got 1 0 0 that means, at the centre of the floor we have got three degrees of freedom two translation and one rotation about a vertical axis. So, corresponding to this degree of freedom that is u 1 we have a ground motion for the second direction here there is no excitation therefore, the third there is no torsional ground acceleration therefore, all this is also set to 0 again for the next floor it becomes 1 for this degree of freedom then 0 0. So, that way for single component earthquake we have this definition of the influence coefficient vector similarly, if there is a two component earthquake then the influence coefficient vector becomes 1 0 0 1 0 0 in the first column and in the second column that is this is I t that. So, in fact, this is actually a column then for the second column we will have 0 1 0 this one indicates the ground motion in the v direction 0 here represent that there is no torsional ground motion. So, for the three dimensional frame we see that we have a I like this next we take a problem of this type in which we have three columns rigid floor this is a asymmetric structure therefore, this is having some eccentricity that is the center of mass the center of rigidity of the floor they would be having some eccentricity. So, this is an asymmetric system since it is a one storied three dimensional frame it has three independent degrees of freedom these three independent degrees of freedom can be chosen in different ways. Here we have considered two different ways of choosing the independent degrees of freedom the model one shows the first way of choosing the degrees of freedom here since we have got three independent degrees of freedom we choose one as this degree of freedom other as this degree of freedom and the third one is this degrees of freedom these three degrees of freedom are the three independent degrees of freedom. In the second model what we have done is that we have taken at the center of mass the degrees of freedom that is one degree of freedom in this direction one degree of freedom in this direction and the third degree of freedom is a torsional degree of freedom about a vertical axis. Again, these are the three different degrees of freedom for this particular model. Now, for these two models, we write down the stiffness matrix, the mass matrix and the influence coefficient vector. So, for the model 1, we see an interesting thing. The key matrix, since it is an asymmetric system, since one column is absent over here, then the eccentricity of the system, it will be having a two way eccentricity. That means, center of mass and center of rigidity will be placed such that there is E x and E y, two eccentricities will come into picture. Therefore, we see that the stiffness matrix of the system is a coupled stiffness matrix. There is a coupling between x, y and theta, not x, y, theta, E 1, E 2 and E 3. All the three degrees of freedom, there is a coupling. Interestingly, the mass matrix is also a coupled one. Generally, we think that if it is a lumped mass system, it will have a diagonal mass matrix. That is only at the diagonals, you will have the masses and of all of diagonal terms would be 0. That is the kind of notion that we have for the all lumped mass systems. However, we will see that in such a situation, where the masses are lumped, but there is a coupling between the inertia forces is corresponding to all the three degrees of freedom or in other words mass matrix over here will not be a diagonal mass matrix. The influence coefficient vector of course, is a simple. Here, the influence coefficient vector is vector which is x double dot g. Therefore, there is no excitation in the U 1 direction, there is no excitation in the U 2 direction, only in the U 3 direction there is a ground acceleration is acting. Therefore, this is 0 0 1 that becomes the influence coefficient vector. Now, let us see how we can obtain these coupled mass matrix. For that, what we will do? I will try to explain to you how the first column of this mass matrix is derived. The other columns are derived in a similar fashion there. So, therefore, I will not explain them. So, let us look at how we derive the first column of this matrix. Now, if we look at this figure, the degree of freedom over here is the first degree of freedom, here we have the second degree of freedom, here we have the third degree of freedom. So, by definition if I wish to find out the stiffness matrix or the inertia force corresponding to the given acceleration, then we release one of the degrees of freedom keeping the other two locked. So, here this degree of freedom is locked and this degree of freedom is locked, then we give a release to this degree of freedom. Now, if we give an unit displacement for this degree of freedom in this direction, then point D moves to D dash perpendicular to line A D for small displacement. Since it moves by unity, here the movement is also unity. So, that this length remains unchanged, the rotation over here becomes equal to 1 divided by this arm length 1 by L. Now, this point if I give a displacement of unity, this will also move in this direction by unity, because there is no restraint provided over here in this direction. So, we do not have any restraint in this direction and therefore, it can move. Now, if it moves by unity in this direction, then this point moves to this point that is B dashed and this movement is again perpendicular to this line. So, that the length this length remain unchanged, this length remain unchanged, because it has rotated by the same amount, this in this direction there is no movement. So, this point satisfies the condition that there is no movement on this direction, but a movement can take place perpendicular to this direction, because there is no restraint in this direction. So, this line now becomes like this. So, the configuration of these rectangle becomes the dotted rectangle after we have given a unit translation over here. Now, in the process the centre the original centre of this square now moves to this point, this movement is again perpendicular to this line. If it moves it has to move perpendicular to this line. So, that there is no change in the length. So, the centre of this moves by 2 o dashed. Since there is a movement in this direction, there is a mass movement of inertia which is developed over here which is equal to M L by 6. Now, note that the mass movement of inertia that is developed is in the opposite direction of the rotation. The rotation is taking place in this direction and the mass movement of inertia is obtained in the opposite direction. The reason is that the acceleration as I told before always acts opposite to the direction of motion. So, therefore the inertia force is acting opposite to the direction. Similarly, since this point has moved in this direction and this centre has moved to this direction, therefore the mass total mass lumped at the centre that moves in this direction. This total other mass moves in this direction or the other mass acts in this direction and this mass acts in this direction. And the value of them is M by 2 and M by 2 because the movement of this point in the horizontal and the vertical direction is equal to half and half if the movement of this point is 1 and 1. So, the inertia force generated here is equal to M by 2 in this direction and the inertia force generated in this direction is M by 2. These inertia forces are again acting opposite to the direction of motion. Now, since the entire system has to be in moment equilibrium, there is a force generated here and here. These forces are M by 6 and M by 6 multiplied by L that gives M L by 6 a moment which is opposing this M L by 6. So, the reaction over here is M by 6 a force here which is generated is M by 6. Then this force and this force, these two forces are now generated. Now, these two forces equilibrate the inertia forces that means this and this makes a couple. These two forces makes a couple and this couple is opposed by this and this force as a couple. So, therefore M by 2 force is generated over here, M by 2 force generated over here so that the entire system is in moment equilibrium. So, we see that if I give a unit displacement over here, this point not only moves in this direction, but has also has to move in this direction by unity. The center moves in this direction that is perpendicular to the diagonal is moves in this direction. The rotational inertia force is developed for moment equilibrium we have these forces generated. Now, if we sum up these forces, then for a unit displacement giving in this direction, the forces generated in the same direction is equal to M by 6 plus M by 2 that becomes equal to 4 M by 6. The force generated in this degree of freedom that is degree of freedom u 2 becomes equal to minus M by 6 because it is moving the force is directed opposite to the direction of the degree of freedom. And the force generated in the third direction is equal to M by 2 or we can write 3 M by 6. So, we have the first diagonal M by 6 has taken as a common and M by 6 has been taken as a common and you have 4 M by 6 for u 1 that is 4 M by 6 and then minus M by 6 is equal to 3 M by 6 that is this one and then 3 M by 6 that is this one. So, these are the three forces inertia forces which are generated given an unit acceleration to the degree of freedom u 1. Similarly, when we give a unit acceleration or unit displacement over here, then for unit acceleration the inertia forces which will be developed here, here and here they form the second column of the matrix. When we will give the same movement unit movement over here keeping these two as locked then the inertia forces that will be developed here, here and here they form the third column of the mass matrix. So, the generation of the mass matrix may require some careful consideration depending upon the problem. So, for this kind of problem you can see that one can generate the mass matrix and this mass matrix is a coupled mass matrix. The P effective would become a M multiplied by i. So, if I multiply M with i then we get this as the that is the third column becomes the effective mass for the model 1. For model 2 it is fairly simple and we can see that since we have declared the degrees of freedom at the center of the mass then for this we have a mass of M. For this direction we have a mass M for the rotational degree of freedom we have got M L by 6 that as the torsional mass or mass movement of inertia about the vertical axis. Other terms are 0. So, this is a classical mass matrix that we generally see in most of the problems. The i influence coefficient vector becomes 0 1 0 because we have declared this as the first degree of freedom, this as the second degree of freedom, this as the third degree of freedom. So, corresponding to the second degree of freedom we have got x double dot g. So, we have got 0 1 0 as the influence coefficient vector and P effective becomes this that means this multiplied by these results in this particular expression for the P effective. The K matrix again we can see that there is no coupling between u and v that is what we expect. But there is a coupling between the this translation and the rotation and this translation and the rotation over here. So, we have this form of the stiffness matrix. So, depending upon the problem we can have different types of modeling, different types of modeling means how we choose the degrees of freedom that is important and depending upon the way we choose the degree of freedom we have a K matrix derived. And a mass matrix and not necessarily that a mass matrix should be a diagonal mass matrix. Again we emphasize this point by showing another example where the mass matrix is not a diagonal mass matrix, but a coupled mass matrix like this. K matrix obviously also will be a coupled stiffness matrix I will be equal to 1 and 0 and the problem is this one. This is a pitch group portal frame this pitch group portal frame we have a mass here, mass here and mass here. These two masses are m and m and this mass over here is m by 2. And one can choose the degrees of freedom in two different ways. For example, this is one way of choosing that is in model 1 we choose the degree of freedom this has one independent degree of freedom and this is another degree of freedom. Here in model 2 we choose the degrees of freedom as u 1 and u 2. So, they have two degrees of freedom because we assume that all the members are inextensible. And in order to maintain the inextensibility we have over here only two degrees of freedom because we can see here that for anyone of them. So, for example, if we wish to give a movement here then automatically this one should move in the horizontal direction otherwise the member or this member will undergo a change in length. So, we can have two degrees of freedom for this model and there are two different models that we can consider. And for these two models the stiffness matrix and the mass matrix would be different. Let us see how we can consider or derive them. We will continue with the same problem in the next lecture.