Jimmy K here, Metal Voice. Look at this. The Metal Voice shirts are now on sale. Just go to the video description to find out on how you can purchase one. Metal! Welcome to the Metal Voice today on the show. Yeah! Don Dawkin, what's going on, Don? Good. Nothing. Working on a new record and I have another record coming out in a couple of weeks called The Lost Tapes. A lot of those songs that never made it onto Dawkin Records. So, you know, I'm just working like all musicians. Not much else you can do with the COVID. You just stay on the right music. Yeah. The Lost Songs, it's 1978 to 1981. It's going to be released on August 28th on Silver Lining Music. Okay, so where did you find these tapes? Where were these magical tapes? Well, I have a house in LA and I have a house in New Mexico. And I decided to move a lot of my stuff from the LA house up here. Okay. And I'd had, you know, like everybody has. You stuff, you stick stuff in the garage and it just disappears for decades.  Yes, yes, yes. You know, so I just said, look, I got all this recorded equipment from the old days when I owned a recording studio, I don't need it. Flight cases from former tours that are broken. So I just got a big truck and just started taking it to the dump to get rid of all this stuff. And I found this box, not a box, like a plastic tub. And I opened it up. It was real heavy and opened it up. And I saw these real, the real tapes, like two inch, which took me back a little bit because nobody records on two inch anymore. You know, everybody's digital now. It's a digital world. So anyway, I was like, what the hell? And some had notes on it and some had no track sheets. I don't know what was on them. But then I did see one tape that said Michael Wagner, Tennessee Tawn Studios, 1979.  So I just figured I have time. We're not, our tour was canceled. A friend of mine had a real, real machine still. We fired it up and cooked the tapes. You have to bake them when they sit that long. And then we just transferred all this music onto a hard drive.  And there was a lot of songs I liked, but, you know, they had a drum machine on it. And I wasn't crazy about that. So I had BJ put real drums on it and John, we do the guitars and so on and so on. So that's how this thing came about. So what did you keep at the end of the day? Did you keep the vocals? Did you keep like from these songs that you found? I guess you're rebuilding it. I kept, I kept all my guitar tracks. John just did the solos over. I redid some of the drums and I kept some of the drums. So I just kept as much as I could, mostly my guitar playing, my bass playing, my guitar playing, my vocals, I kept all that. But there was two or three songs that had like a real cheesy drum machine on it and it just didn't sound very good. So I thought I could leave it like that, but it sounds like hell. So I just had BJ put, you know, better punchy drums on them. And then the guitar solos, I had John do or we do it. Well, I had John do solos because a couple of songs didn't have, didn't have a solo, just blank. You know, it's been bothering me since 1982. Did you find any sort of scratch tracks with you with the scorpions on the blackout days? I've always wanted to hear that. I've been asked that a lot. Yeah, I have some. But I would never give it to anybody or play it for anybody or let it out of my hands. Because I remember when they were re-releasing blackout, I'm not trying to change the subject. But I remember Matthias Jobs, I interviewed him and I asked him on the deluxe version that they were coming out with the 40-year anniversary. Will they include Don Dawkins, maybe a few tracks from Don, you know, back in the day? And he didn't know. So I'm trying to find somebody who knows. Well, that's not an answerable question. I didn't sing any lead vocals. I just sang background vocals. That's it. But did you do the scratch tracks sort of like as they were building the album? A couple, only two songs I scratched. Okay. Just two. All right. But, you know, I'm sure those are long gone. I mean, why there'd be no reason for them to put me on a song that Klaus sang, you know. It was just a scratch track just to save Klaus's voice because you'd had some vocal problems. But I didn't sing, I only sang like a couple songs, like a really quick 20-minute scratch track just to get the melody on tape for Klaus. To me, I got to ask you this question because it's been bothering me for like almost 35 years. How long were you in the studio with the Scorpions? Well, I mean, as far as me participating on the record, just like two days. Okay. You know, but as far as me being in the studio with them, I was in there for a month because they were making blackout at the same time I was making break in the chains. Yeah, yeah. In the same, in the same studio and except it was in there making Russian Roulette. It was except Scorpions and me and, you know, they had three different recording rooms and we were all in there.  He had like a Dieter Dirks mom, Frow Dirks. She would cook us dinner and we had like a little cafeteria and we'd all take breaks or go out to dinner down to the local pub called Peters and, you know, terrorized and we all hung out for a month. And we're playing video games and playing Galaxian and Pac-Man. And, you know, so it was about a month I was there hanging out with those guys. Wow, sounds pretty cool. Nice days. How about any. Okay. So going back to your, your unreleased tracks that you're going to be releasing the lost tapes. Are there any songs from there that you're saying, you know what, maybe I'll hold on to this one. It's a gem of a song for another docking album. An album. No. Okay. I mean, songs. Look, every every musician, usually when you're right, you got to remember back in the days of records, you only had a certain amount of time to put music. I think it was 33 minutes. That's it. You couldn't put any more than 33 minutes on an LP. So, you know, when the days of the CDs came out and like Metallica puts out 15, 16 songs, you know, you had a, when it went digital, you could put more songs. But we'd always write 14 or 15 songs and, you know, and just pick the ones we thought were the best ones. Okay. And sometimes I write a song, I've been making a demo and I have a little drum machine. I'm in my little apartment playing guitar. I'm singing. And I just, you know, it would work on a couple of days and go, you know, this song's just not happening. I'm not down with it. So I just abandoned it. Yeah. Okay. And congratulations for winning the, at the metal Hall of Fame, the award. That's where I actually met you. I just got out of a hospital and I was a little just loopy still. So, but yes, it was a great honor. I appreciate it. It was very kind, really happy, you know, so it was great. And from what I remember is, and, you know, if you want to talk about this great, your health, I remember your hand was blown up.  It was blown up, you know, you can barely move it. I mean, how's that coming along? Unfortunately, it's about the same. Paralyzed? Yeah. My, my hand and my forearm and my triceps are still paralyzed. It's been seven months. You know, the doctors just kept saying, be patient. It'll get better. It'll get better. I can move my fingers about another inch, but that's, and my thumb doesn't move at all. And my four fingers are still blown up and swollen and look like sausages. And, you know, so it's been seven months and it's not getting any better. So I have to accept it. They say it might take a year or a year and a half. And all I can do is be patient and pray that I wake up one morning and it's starting to work. But the reality is it's not getting any better. So I might, my guitar playing days might be over with. Wow. That's pretty sad. But I played for 50 years, 50 years. I got to play guitar. Yeah. Yeah. But at least, yeah, at least it didn't paralyze me all the way. Yeah. So it could have been worse. It could have killed me. So it's not that bad. Don't talk to me, please.   I guess that was after the surgery, correct? Oh yeah. I fell in the snow after the surgery. That infected me. Yeah. I went to open the door in my car. It was snowing and I, and I tried to use my right hand, which was paralyzed, you know, but it's just habit. You know, I'm right handed.   So if I would have, if I would have fell face down on that, that rebar and railroad ties, you know, I, we wouldn't be having this conversation right now. Yeah. So, so tell me, and this is the last question on your health, where did it all begin? It was just, it just all of a sudden you woke up one day and your back paralyzed arm. Is that how, what happened? No, no, it was just like little things over a couple of years. I was riding my Harley. I kept, you know, taking it back to the shop saying there's something wrong with my clutch. It's really hard to pull in, you know, the clutch is very stiff. And they said, no, it's fine. And then when I ride, you know, I was having a hard time holding the throttle like my hand would get weak.  I pick up a cup or I pick up a plate. Let's say the dinner and my right hand. And it felt like it weighed like 10 pounds. And I'm like, what the hell's going on? So then I started getting a tingling sensation in both my arms, like my fingers would go into sleep. So I went and got an MRI. And they did a full spinal scan and they said your spinal cord is completely like crushed compressed. And that's, and you're getting blood.   And they asked me, you know, did you get an accident?  I said, no. Did you take a bad fall? I said, no. They said, we don't see this very often. Down five, six and seven, which is the place when people break their necks, they usually die. And I said, no, I don't, I don't remember having any bad falls or any accidents. I don't know what happened.  So the bottom line is they said, you can wait. But the longer you wait, the more nerves that will die. And the nerves will not come back. And at some point you will be paralyzed. So I said, okay. So I went and found, thought I found the best doctor in LA, Cedar Sinai, and had the surgery. And it didn't go so well.  So it's where it goes. I just drew the short straw, man. Yeah. You, you won the wrong lottery. That's when my doctor tells me, hey, Jim, you won the wrong lottery. Yeah. Yeah. I mean, I mean, the whole point was so I get my hands back. Yeah. I didn't think my, and then not only my hand, my forearm, my, my whole right arm, I could move it now. I can lift it up and down and move my fingers a little bit. But, you know, I can't pick up a pen. It's on my name or I can't hold a fork or. So I'm learning how to be a lefty. I'm just, I can't ride my Harley. I want to ride my Harley. Yeah. Yeah. Well, I mean music. Soft stations. Yeah. Well, I mean, there's always music, right? I mean, what's, so what's the plan for the next studio docking album? We're talking about the, the law songs, but what about new docking? What's going on there? Well, same thing.  Yeah. We wrote like 17 songs for broken bones. Nice. So, so many of those songs didn't make it onto the record. So I went and looked into my, dug out all my old hard drives and John did. And John's a great guitar player. He had a ton of ideas. I've had a ton of ideas. And so John's just doing the guitar work and I'm doing the lyrics. And there were actually, I found like three songs I played guitar on. We're going to keep for the record. And that's, we're working over the internet like a lot of people. You get on zoom or Skype and you know, John has a studio in LA and I have a studio up here. And we just exchange ideas and they say, what do you got? And he goes, what do you got? And you know, he'll play me a song. I did also hang on a second and all because I can't type anymore with my hand. I bought, you know, vocal programs. You just talk into a mic and it types for you. So I'll just start typing out lyrics. And that's how we've been working. And we're about half done with the record. We've got about seven songs so far. So what do you, what are you looking at as a timeline in the next six months? Um, that's a hard call in the perfect world. You know, we were going to have an album out this summer and then Corona hit. Yeah. And that changed the world for everybody. Yeah. So it was a good timing to put out the lost tapes as a buffer. A buffer for our hardcore docking fans to have some new product from us because it'd been so many years. And then I was, we were talking about, we're thinking about re-releasing broken bones because I never thought that that album got a fair shot. It was a really good record. I was really proud of that record. But the record company we were on at the time didn't do anything. They didn't promote it. They didn't advertise it. It just kind of came and went. So silver lining might be re-releasing broken bones. And, um, and I had an idea, which I have, I'm thinking about releasing a best of docking ballads for the holidays like Christmas time. It's never been done. We never, a lot of bands do that.   So let's just put out a whole album of ballads for like Christmas. And I think that would be real cool. So I'm working on that too right now. So over, over time, docking has evolved. I mean, what's the next, is the musical direction going to be the same next time around? Or is it just, it's going to evolve into something else? We're going retro. We're kind of going retro. Okay.  The world changes. You're right with your right. You know, you're influenced by new bands and other music. But, you know, I know where my bread is buttered as far as the classic docking sound and John's real good at doing that. And so I said, I think we should write a record between the lines of tooth and nail and under walking key. And that's kind of the direction I'm writing in right now. All right. And how about the classic band reunion went off shows? No, no, mixed retired, mixed retired. Yeah. Yeah. All right. Doesn't play an important mix completely retired. George has Lynch mob. I've done a few shows with George. We were supposed to be playing. We were supposed to do 10 shows with George this summer. Yes. But the COVID, the COVID hit. He doesn't fly. He doesn't want to travel. And so that all went out the window. All right. Well, there you have it docking the law songs in 1978 to 1981 on August 28th on silver lining music. Thank you so much. You can get on vinyl too. Yeah. You can get on by coming out on vinyl. Nice. So CD vinyl and cassette. No more. Was docking ever released on an eight track the first album? No. No, we just missed that era. You know, I have a few platinum records on my wall that have a cassette in it underneath. Okay. I love my tracks. I remember listening to Y and T in my car on an eight track, you know, and then I had a in high school. I had it sold 63 Corvette and I had a little four track. And then and when they invented the eight track, I thought that was so amazing. Well, thank you so much. All right.


It's all about like power through, push through, no pain, no gain. But at a certain point, you have to actually start recognizing that your body has limits. And that can be a really painful experience. This past weekend, I was supposed to take my dogs up into like the wilderness of this cabin for like a couple of days. It was gonna be great. We were gonna disconnect. But as soon as I got up there, I realized that I, I'm really not capable of walking these two dogs that I'll weigh me by 50 pounds put together. Like on uneven services by myself. And it was just me out there and my leg was getting really painful. And I made the decision to call it quits, come home a day early. The part that sucks about that is not missing a couple of days of vacation. It's recognizing that I actually do have limits and I can't do everything. And that makes me really sad. But I've also learned to recognize that there's a lot of beauty in it. And I'm really glad I made that call so I could protect my body, not put myself in danger, not push things too far. So we came back and had a sleepy nappy kind of puppy weekend instead.


A fast and robust magnon transport protocol is developed in a one-dimensional spin chain by exploiting an approximate mapping with a harmonically trapped particle and adopting known analytic control protocols, resulting in faster and more stable transport compared to finite time adiabatic protocols, with a sharp transition between effective and breaking down regions. This article was authored by Anthony Kealy and Steve Campbell.


To me, I want to catch that nose and keep that shoulder stood up and then when I open it kind of comes through, it's like riding a bicycle I guess, you know, you've got to keep this stood up and I want to get that nose there to where it's soft and then let it come through. So later on as we all get going, all I've got to do is just grab it there and everything's going to come through with it.  So I mean a lot of it, like on them younger horses, now I've got a little correction on this horse, a lot on them two-year-olds, I'm going to have a snaffle and I'm going to pull back here a lot, I'm going to pull to my hip, drop to my knee, like so when I trot them around here, I'll leave and do it now because I think this is a really good exercise.  Change this way, pull it to your hip, drop it to your knee, pull it to your hip, drop it to your knee. You know what I mean, just to wear that horse and now he's wanting to trot around and he's staying behind the bridle to where, when he's behind, when he's bridled up and soft like that, he has everything through here soft. He's back, he's back is around, he can get, okay, so this horse is being pretty good right here, so I'm just going to kind of let him have that care, let him work it there, he jumped off a little bit, I'm just going to draw him back. Like I said, I want to create them to want to do it, only when he gets in trouble am I going to help him. Right there, he kind of bounced that stop a little bit, just going to forget it even happened, because he's done a lot more good moves.


There goes the TV there goes the TV There it goes worst mistake I could ever have met But oh God is it gonna be smashed is it gonna be smashed?  Oh, thank God. Oh, no


I admit it, I was wrong. This is the best desktop diode laser you can buy. My name's Jim and this is the Edge of Tech. So recently, about three months ago, I did a video on the Artur Laser Master II. I really liked that machine, I used it a ton and I thought at that time, it was the best desktop diode laser you could buy.  This is the best desktop diode laser you can buy and as we walk through the specs today, I'm gonna tell you why. Let's go. I just wanna interject quick and tell you this video has been brought to you by Thangs. Thangs is quickly becoming the biggest 3D model community out there. With over three million models on their site, you can find almost anything and if you don't have it, you can put it on Thangs anytime you want.  Click the link in the description below to get the mini-gym by Wexter and that'll take you straight to Thangs. Thanks Thangs for sponsoring this video. The first thing I wanna talk about is the board. Now it sits in the front left of the machine behind this panel right here and it's a brand new board for the Laser Master II Pro. They say that it's a 32-bit board and it can increase the speeds from 4,000 millimeters a minute to 10,000 millimeters a minute with the brand new board. Has a new algorithm that is motion optimized so it's supposed to be a lot faster and it allows many more safety features built in on this machine because of the board. The board is also 24 volt and the power goes right here. As you can see, there's a USB here and a cable right here for the standalone controller if you were to have one. Now I don't have one of those, but they're pretty cool. When they start being more prominent, I'm gonna get one and I'll do a video on that later. Talking about safety features, the first one you can see right away is this emergency stop button. You push it down if you need to. It'll cut the power to the laser and it won't keep firing. So that's a good way to go with that emergency stop button. Remember to release it and you just turn it and let it pop back up. You wanna make sure it's up before you try to do any burns or it will not work.  The other cool thing is if you're burning and maybe you tilt the machine or the machine moves at all like this, it will detect that and shut the laser down as well so that's another great safety feature. The other cool thing is maybe your laser is burning like this and it stops and it stops for too long and it's in a stationary position. It'll actually cut the laser diode as well. That way it doesn't just keep burning down. So if this were to somehow get stuck, it'll actually cut out the laser and that's awesome. This right here is the sound and light alarm so if it does detect a flame or anything like that, it'll buzz this alarm here, a light'll come on, it'll make a noise. Now I haven't tested that, I should probably just throw a flame or some sort behind it and see what happens but they say that's what happens, it'll come on, it'll make a noise and that'll tell you that there's a flame detected or something's wrong with the laser as well. The other cool thing is if somehow your USB here gets pulled out or stops from your computer, it'll actually stop the laser as well so it won't keep going, it won't freeze the laser where it is like some lasers do, so that's really good. It detects if that USB gets pulled out. That only works if you're not using the offline controller here too. The other cool thing is this laser safety shield right here so this is down so you don't get exposed to most of the beam. Now a lot of times your material will actually sit way up in here so it'll be pretty close to the material and it's a lot less chance of you walking in the room and accidentally seeing this if you don't have it in an enclosure. Now I always burn with mine in an enclosure but if you don't or you don't have one yet at least you have this protecting you. Now that does loosen and flips up like that if you need it to and that will help you with the focal distance but if you need to lift that up put your material under there and everything then slide this back down and lock it in. Make sure it's tight so it doesn't just flip up.  So this safety glass is a pretty awesome thing to have around the edge of the laser. That was a little about safety. Now let's talk functionality. This right here came with mine. Now I believe this is an add-on. I'm not sure if it comes with all of them or not but I love the Z adjust here so because this is a fixed focus laser you never have to turn the focusing down here. You do have to adjust it up and down to get it to work correctly. Now the one that comes with it standard I believe you slide up and down by loosening a screw or a bolt but the one that came with mine I added on and you just twist the knob here and it'll go up and down to the focal length that you need. So let's talk about focusing while we're talking about this shield. The first thing I would do to focus is lift up my shield and then I would lock it in place just a little bit so it doesn't fall. Next I would grab the included bar right here. Now this is for focusing and I would set it right on top of the material that I wanna burn. In this case we're gonna use this cork coaster. Then I would turn my adjustment so it goes down and just touches just like that. So you want it just to touch the housing just like that. Then you can slide this out, you can flip your housing down and then lock it in position from both sides. Now you're ready to go. That's at the optimal distance here for the laser to burn this coaster and that took all of maybe 10 seconds to do. It's really fast to focus these now and that's a really cool thing for these lasers because it used to be a pain in the butt to dial that in and get it all focused. The next thing I really like about this is it actually has physical end stops. Now there's one here for the Y and there's one here for your X and I tell you what these things really help out. I've been testing some without end stops and I tell you what I am spoiled because this one does have the end stops just like the laser master two did. One of my favorite things that this thing came with is probably this cable train here. I absolutely love this. All the cables get hidden in here and it really looks nice. You know, it's free flowing and moves back and forth as needed and I tell you what, I really, really like that feature. I'm glad they did this. They did not do one up here for your X but for the Y they did it and it looks great. There's still a little bit of room in there if you wanted to run an airline or anything like that but I tell you, I really love this cable chain. Something else they did is they got rid of a lot of the weight here on the X cables and I tell you, this is really light. It hasn't gotten in the way of my stuff yet but you could easily put something a little bit thicker in here to help it stay up if you really wanted to or even print another cable chain if that's what you wanna do but I haven't had any issues. It's super light which keeps the head of your laser very light as well. Something else I really like is that it has the measurements across your X and Y axes so you can see them there and if I come over here you can actually see them. It goes all the way down the back and it actually has them on the back as well. That's a really cool thing if you're trying to get exact coordinates or measurements but I tell you what, I usually burn the grid on the wasteboard anyway and that really helps me out but it is cool to have them and in my case it's actually in inches. You can see it right down there in inches. Another great thing they did was add the second motor here on your X axis. This drives the X gantry on the Y axis back and forth like this. Now the reason why this is awesome is because it helps take some of the weight off the motor on the left side so it does connect with a bar here to keep everything in line. There is a belt on both sides as well but this right here will help speed up this laser just like they say it's gonna and I tell you this thing can move much faster when it's two motors instead of just one trying to pull everything back and forth. I really like that feature. Another great reason why this is the pro. Another cool feature is this holder right here. This holds the bar that helps you focus the laser here so you pull this out and it just sets in there so it's not just rolling around loose, you don't lose it, you always can keep it right in the front. Something else I wanna talk about is how structurally sound this is. You can see that they actually include inside corner bracing so you put that in when you're building it along with being bolted together from the outside as well. So this thing is very sturdy. It holds up to the speeds you put it to and I really like the frame of this machine. All right, real quick, I wanna show you some things that I've been playing with on this laser. This right here is a lion I found on Google and I really like this picture. So I threw it on some colored scratch paper. As you can see, the rainbow scratch paper came out really, really nice. I did not clear coat these so I just wanted to show you exactly how they came off of the laser. You can see it came out really good. The detail is absolutely phenomenal in these and I think that came out very good. Now while I was dialing it in, I was playing with different speeds and doing tests like you should and I did one of our signs here. It says even Jedi's take naps and just kind of a fun thing there but I was trying to dial it in with this one. Just seeing where my setting should be before I did that lion. When I got that done, I started looking into this right here. Now this is gold scratch paper and it's not terrible. I could done better. So I'm gonna keep working on the settings on this one. You can see this one really shiny right there. So I should be able to get this much, much more shiny. Then I went on to tiles and I did one of my magic numbers tiles. I'll link that above. And it came out really nice. As you can see, 15 in my opinion was the best. I really liked 15. I believe this could go much faster than 1200 millimeters a minute. So I'm gonna adjust one and try to get a new file out there for this because why not see how fast this thing can go and produce some nice prints if we can, right? So once I did that, I moved on to this one right here. Now this was a reverse tile. I put black down first and then red on top of it. And you can tell I was way off center on this but it's really kind of fun sunrise here. So you can see the red in here. It burned down through the black into the white of the tile and it left some of that black for the clouding and stuff like that. The reflection looks really nice. It's supposed to be over a lake or something. And I think it came out not bad. Again, I did not clear coat these so they're not gonna be as shiny as I normally would be but I just wanted to show you how they came straight off of the laser. The next thing I did was the Nicky Norton test. I had to crank down my ISO here to even show you this because it's white but this is the next thing I did. Now this is a Nicky Norton test. I'll put the links in the description for the tests that I do. This is the Norton method tile test pattern, 100% max. So it starts with a percentages 10 all the way up to 100 and 250 up to 3000. Now I did not modify this for the LaserMaster 2 Pro because we're supposed to be going up to 10,000. So I need to do that on this one but I had this file and I wanted to run it. So this will tell you where you should be in the white tile method. Once I got done with that, I ran this one right here and this is just a nice scene with some elephants. This is the Nicky Norton white tile method. It is not bad. I could do a little bit better, I believe. Actually, I'm gonna dial these settings in better so you're gonna get some more definition in the blacks and stuff. But I really love the details, the clouds, the grass, everything you can see in this. And this is just the settings I ran right from that test tile. So I definitely can dial this in better and I definitely will do that. But it wasn't too bad. I just wanted to show you some tests that I did do with the laser.  I will pop those on my social media, so stay tuned. But overall, I really like this laser. So there you have it. All of those specs combined, the safety features, the new updated parts, everything combined is why I changed my mind. And this, in my opinion, is the new best desktop-dialed laser you can buy. I highly suggest you check one out. They're pretty awesome. They're always upgrading the firmwares. As they move forward, I've done a couple firmware upgrades since I got this beta machine. But I tell you what, it is definitely worth it. Now, this machine was sent to me free from X-Banks. You can find a link in the description below. I really appreciate that. I have been testing it for quite a while, a couple months now, and I really think this is the way to go if you're looking for a desktop-dialed laser that can do just a ton of stuff. Well, I hope you guys learned something today and as always, keep burning. Hey everybody, give me that thumbs up if you liked the video. If you haven't already yet, please hit that subscribe button right in the center. And as always, click the little bell right over here if you wanna get notified anytime we go live or put out another great video right here on the edge of tech.


Bro peppy if you let him go to bought a edge right there. I swear to God, dude. I'm over you It's just a belly peppy. What are we doing?  I didn't want to play him go to bear free and cold Free and cold dude. I mean the Raiders gotta be the most disappointing team in the league I would thought the Raiders were gonna be like Correct, I mean that whole division we thought was gonna be the NFC East You think the jets are better than Giants who y'all like more jets Giants who y'all like y'all hear this dude on the mic I'm gonna have As much as sweaty like this much as that I'm playing Okay, I can't bail off the right side right there. I feel like my tackle was getting flipped Sorry I kind of nervous. This is not to be fired, bro K max empty. Oh Step on him then Calvin Let's go dot kid Prime. Are you drunk? Bro tell me I need you to strip. I need you to strip not hit him on these are the strip Just send five every play W. It's just nasty. We just did bro Jesus Christ.  I don't know why y'all had it like a losing. He got mine. It might be him. Ah Don't know why I thought I get a better throw right there, bro I don't know why man like like. Oh my god, dude Like bro. This is so dog man. Oh my god. Dion's has got so pumped. I gotta switch the on and pet people I really do Jesus Christ right now, dude.   It's chat is the worst, bro I like brother the dude literally set free safety the whole game, dude. I'm trying to free safety Y'all be looking at God dude to running back passes or his entire points, dude Oh, the Andre Hopkins is just such a difference maker, dude. Oh my god Yeah, he would have definitely won the whole game. He would have played defense special teams offense.    I'm just gonna bro. This is miserable to play like this is and we just break a tackle and oh my god Bro deans on the ground as pet Pete those are just on the ground dog Happy if you let him go to bottle edge right there. I swear to God, dude. I'm over you It's just a batting pet pee. What are we doing?  It's just that good a door angle route, huh? Just no way you're kicking this girl. What do you expect to do?  I don't know why he's not playing aggressive on his angle route to the running back He's just like so he's not even like getting beat on the row.  You took bro, but don't worry I'm gonna go for a touchdown. You know why cuz I'm a man, you know, I'm 40 Nothing pet man pet Pete. It's just like he's not I don't know explain what's going on Like he's not getting they not going at the pet Pete, bro It's like I don't even I'm not gonna buy a corner if he guards them. You're saying like he's not the guy I'm gonna put he not the guy. I'm putting on the running back, you know bad play wreck facts strip stack. Oh What night train apex? That's what's crazy. I had night train for two months. No train is that no train is that I take my the lineup. Y'all tell me he good now. Nuts Nuts, bro. Ashley. No, GG. Don't you Jesus?


Hey everyone, my name is Vincent and I will be speaking today about a topic that I find very, very interesting as a product manager, which is making good decisions as a PM. This is a very important topic to discuss as a product manager because you'll often hear that PMs are at the intersection of the customer, engineering, and design. So which means you'll be making a lot of decisions. I think this is a very interesting topic for a lot of you who are aspiring to become PMs or already PMs, but I do want to highlight that making good decisions is important for everybody in every industry in any discipline. For example, if you are a chef or a firefighter or an architect, an airplane pilot, you'll be making decisions. So I think this framework will be very useful for many, many people. Today the topics that we will be discussing over the next 20 to 30 minutes will be decision making. First, I will give off an introduction of myself. Then we will move into why decision making is important as a product manager. We will cover some of the main problems that I see product managers face today as it relates to decision making and then how we can use some tools and frameworks to make decision making more accessible and more logical for product managers. And last but not least, I'm going to go over my favorite framework for decision making and go through an example with all of you. This is a slide a little bit about me. My name is Vincent. I'm based in Toronto in Canada. If some of you don't know where Toronto is, it's just north of New York City in Ontario, in the province of Ontario. I moved to Toronto when I was very young from Hong Kong and China and I grew up studying accounting and finance at York University and then I did an exchange semester in Singapore where I studied marketing and strategy. My career has spanned over a decade now working in several different functions. I was an accountant. I was in M&A. I was in venture capital. I was in product marketing and finally I am now in product management. And my career has touched on various different industries. So I've worked in banking, in payments, in startup, in B2C and in B2B SAS. And product management is somewhere where I landed after 10 to 12 years of kind of finding my way around different places and seeing what was interesting to me. So this line on the left of the screen is kind of how I see my career. It just wasn't linear at all. And I found product management through some trial and error. On the personal side, I love traveling. If it wasn't this day and age, I would still be traveling a lot. I love seeing mountains. I love deserts. And since it's COVID right now, I've found passion in other things. I love reading. And so if you'll often find me reading a book about entrepreneurship, about technology, about, you know, self-help books, about decision-making, I love to eat. I'm sure a lot of you will agree with me that food has been a great place to find an interest these days. I like video editing, very amateur about it.  So not only am I a product manager, but I'm also a husband and a father. Great, let's go into the next part of this presentation about why decision-making is important. So product managers sit at the intersection of engineering, design, the customer, marketing, the overall business. Decision-making is at the very core of the product management function. Let's go over some areas that PMs have to make these decisions. The first one is, of course, the product. What is the product? What is the strategy that you're going after in the next ten years, five years, three years, one year? What do we do first and what do we do second? Next, you have to work with your design counterparts to understand what does the customer want and then design the user experience. Should the customer go straight to step two or they go with the step three, go to step one? What does that overall end-to-end experience look like for the customer? Next, you have to work with engineering. So PMs will often be deciding with the engineers. What is that version one of that feature, version two, version three? What is our MVP, which we commonly called it, and then how do we build it? Next, you have to work with marketing and sales to decide if your product or feature is launched to a certain target audience. Who is the target audience? What is the demographic? What is the way that we communicate our value proposition? Et cetera, et cetera. Aside from the business, PMs are often making decisions for the entire organizations, something we call organizational design. Do we need more designers or do we need more engineers? How should we structure the team? Do we have a six to eight ratio from engineers to product managers? Do we need one product marketing manager or two designers? Organizational design is something that PMs will often have to influence as well. Needless to say, decision-making is a very critical part of being a product manager because it helps make your delivery team very, very efficient and very, very effective. I think if you look at various product management books and resources, you'll see that PMs not only have a delivery deliverable, but you also have to make sure that your team is unblocked. And what does unblock mean? It means that you as a product manager need to make decisions fast enough and good enough so that your design, your engineering, your marketing, and sales teams can go do their job. Let's give you an example here. Let's say that as a product manager, you need to design a feature A.  As a product manager, you need to understand all the facets of that particular product and the strategy so that you can work with your designer so that by the time engineering is ready, the design is also ready so that everybody is passing the baton to the next person very quickly, very quickly. That is a very important part of being a product manager, unblocking your team. Let's talk about what should a good product manager do. A PM should be competent in multiple facets. The first one is navigating complexity. Product managers are involved so early in the product development cycle so you'll find yourself often in ambiguous situations. The problems that PMs have to navigate are often very shrouded and clouded in a fog of war. It's important for a product manager to kind of cut through the ambiguous situations to make sense of it all. For example, if somebody, if a customer comes to you and says, I need this, as a product manager, you need to understand what exactly is that customer looking for, dissect it and communicate that back to an engineering and design team that did not directly talk to the customer. So it's very important for a product manager to see through with clarity and transparency what a customer actually is looking for, prioritize whether that's important and then communicate that. Next, a good product manager needs to proactively plan. And what does that mean? Proactively planning means kind of like what I mentioned in the previous slide about making sure that your delivery team is unblocked. You need to kind of see three steps or two steps ahead. You need to look at the problems, assess the prioritization of certain things and decide what should go first. For example, let's say you talk to a customer and they say, I need three things. Is the priority in which you're designing engineering team doing these three features? Is it one, two, three, two, three, one or three, two, one? Which goes first? If you choose one over the other, what are you trading off? A product manager needs to make a decision on prioritizing and the only way to do that is understanding the big picture and the strategy and then planning the sequencing in which your delivery team will move forward. Next, of course, as a product manager, you need to de-risk delivering. Product managers at the end of the day responsible for delivering results, delivering features to deliver outcomes. This ties back to the earlier theme that we talked about, about unblocking teams. As a product manager, you need to make sure that your engineering and your design teams are moving quickly, like a machine. To make sure that they can deliver quickly, you need to make decisions fast enough and good enough that they can proceed and deliver on time within budget. Let's say that your executive team or your stakeholders expect you to deliver a certain feature by June of next year. If your team is trailing behind, you're not making the right decisions, your June delivery date may stretch to July to August to the end of the year and that may have a ripple effect throughout the organization that could impact business results like revenue and active user growth. It's very important for a product manager to be good at making decisions that keep delivery on schedule. Next, a product manager needs to be very, very clear. A product manager is talking to all these disciplines, engineering, design, and the customer and it's important and critical for a product manager to understand the opinions of all these stakeholders so that you can communicate back to each of these other stakeholders who are not necessarily communicating understanding all aspects of the business problem so that they understand why. Why is one of the most important things that a product manager can provide clarity on? Why is going back to the motivation in which a business needs to achieve a certain objective? For example, this pair of glasses I have, if you're working with engineering design to design this pair of glasses, it's very important for a product manager to communicate to the engineering design teams of why these glasses are designed a certain way. You can't just tell them design glasses that are circular, can attach to ears. It's critical for a product manager to talk to a customer and tell the engineering design teams why glasses have to be designed a certain way so that they understand the problem in which the customer is facing so that they can go design it properly. This communication will allow for a more complete and coherent team that's working towards a common goal and a product manager is the glue that helps all the stakeholders work together nicely.  A product manager needs to make sure everybody in the organization is working coherently, working happily and very, very efficiently. When decisions are made, oftentimes the product manager is making that decision. It's the job of the product manager to get the perspectives of all these teams, understand the different opinions and data and the trade-offs before making that final decision and communicating that decision to everybody. Not everybody is going to be happy or agree with the decision you made, but they should understand why you made that relationship and the PM is the center of that, where you need to manage the expectations and the stakeholder relationship so everybody at least understands why a decision is made. Next, we're going to discuss what is not okay in decision-making today. These problems I've seen very commonly face product managers across the board, either early in their career or later they're in their career. These are the problems that frameworks and decision-making tools can help you fix. The first four areas that I see product managers face are poorly defining a problem, not having enough data or decision principles to make a decision, ignoring stakeholder opinions and kind of jumbling the options. Let's go into each of these. Poorly defining the problem. When you need to make a decision, the decision needs to be hinging on a core problem set. If the problem is not clearly defined, it is very difficult to find a solution. We're going to cover this in example in the next slide so we can hold off on this one. Lacking decision criteria and data. This one is probably less common, but when product managers and any stakeholder make decisions, they need to make sure that it is founded on certain principles. Kind of like a pros and cons criteria set. Is this criteria more important than another criteria? When you communicate to stakeholders around the decision you made, they will ask you how did you decide on this decision based on this criteria? It's important to lay out the criteria for your stakeholders so they understand how you got to that answer. Ignoring broader perspectives is another common problem where I see product managers not soliciting a wide divergent opinion base. This gets into multiple, several following problems. For example, people feeling left out of the decision, people getting blindsided, or the product manager not having enough details and data to actually assess the options. It's important to solicit opinions broad and wide so that the product manager can consider engineering, marketing, sales, data science, design considerations before coming up with the final decision. Last large problem that I see with decision making today is jumbling options. What does that mean? Decisions have to be made based on assessing multiple options that are available. For example, if you have to make a decision on whether a certain feature needs to look a certain way, you need to have option 1, option 2, option 3, and option 4. All of these options need to have pros and cons. They need to be distinct from each other. Enough that decision 1 is different enough from the second one. We can cover that in the next couple of slides as well. Some of the other problems that I see with decision making today is defaulting to the highest paid person in the room and executive in the room, who may not be close to enough to the problem, but just by mere fact that they are an executive or a senior member of the company, they make a decision. This is very, very wrong because it impacts morale above all else, but this executive may not be even close enough to the problem to understanding the trade-offs. We can, wanting consensus is another problem that I see where there is a reluctance to make a decision because not everybody in the room is happy with a certain decision. This bullet particular I've seen in many, many companies outside of software and incumbent companies whereby everybody sits in a room and out of 10 people, the decision is not made because all 10 are not voting yes to a certain decision. This is incorrect because it slows down decision making, which impacts delivery. This makes it too long to make a decision or actually not making a decision when you leave the room. These problems will be eventually solved with the framework that I will outline next. The framework that I want to talk about today is the Spade Framework, a very common framework that we see every single day at Square and companies like Amazon. This Spade Framework, the S, P, A, D and E is an acronym that stands for Setting, People, Alternatives, Decide and Explain. We're going to go over each of these topics in the next couple of slides with an example, but you'll see that it's very simple. It seems very obvious, and it encompasses a lot of other frameworks that you might be very familiar with during decision making. This is one of the frameworks that I'm recommending, but by no means is it the Bible for decision making. This is just the one that I want to talk about today, but please feel free to use the one that you're most familiar with. Let's go into an example. S stands for Setting. What does Setting mean? Setting is helping the reader understand what is the problem that we're solving. That's context. Somebody is reading this decision one week from now, one year from now. They understand, I see, this is the context and situation in which this decision was made. It also tells the reader why is this problem important? Why are we deciding to look into a solution for this problem now? Imagine setting as the preamble or the introduction to the problem. Let's go into an example. Imagine that you are a product manager for a video streaming service. I'm not going to name any particular service, but just imagine the video services that you did. This video streaming service has a membership or monthly fee plan, and if you sign up for the membership plan, you get to skip ads. As a product manager, your goal is to increase the number of subscribers to this monthly subscription. That is the problem that we're trying to solve. We're trying to increase the number of subscribers to the membership program. The specific objective that we want to achieve with this decision and what we're trying to weigh is to optimize for net news subscriber growth through ad placement. I've highlighted in green here the key points here. The problem that we're trying to solve for is increasing number of subscribers. Why is this important? It is important because we want to optimize for new subscriber growth through ad placement. This is a very good setting because it's really a problem what we're optimizing for and what we're trying to do. Let's look at the flip side. What is a bad problem statement? A bad problem statement is we want to increase revenue but also want to reduce turn. We can do this by either increasing price, adding subscribers, or finding new ways to engage users. This is a poor problem statement because it does not define the core problem that we're trying to solve. We're trying to solve too many things. In the first sentence, we are both trying to increase revenue but reduce turn. Why is this a bad problem statement? It's because we're trying to do too many things. Which one are we trying to do? The next sentence says that we can either do this by increasing prices, adding subscribers, or finding new ways to engage users. Again, there's too many problems in here. Increasing a price, adding subscribers, and engaging users those are all different problems with different stakeholders with different data, different analytics, and different possible solutions. This is a bad problem statement because we're trying to do too much. We also don't know why this is important. I want everybody to focus on, obviously, the good example here. The bad example is just for you to see how we could make a bad problem statement very easily by jumbling too many things into one sentence. Let's move to people. P stands for people and people is defined as the core working group that makes the decisions. We can use a simple RACI for deciding who the people are that are involved in this decision. R stands for responsible. The responsible person is the person that gathers the information and also makes the decision and recommendation. This is often the product manager. A in the RACI stands for approver. In the approver, you can imagine as the person who is likely more senior than the responsible person and has the ultimate ability to veto the decision. Let's say the product manager is responsible and makes a decision and the next person that they report to, let's say the head of product is the approver. That head of product needs to have the ability to ultimately veto the decision. If somebody more senior than that person or the CEO can veto this decision from the head of product for the person who's approving, we have to switch the approver. The approver must be the person who has the final decision to say no or yes to the ultimate decision. C stands for consulted. Consulted is a group of people. It can be many people that provides data information and opinion so that the person responsible has enough information on hand to make a final decision. The consulted can be design. It could be engineering. It could be data science. It could be sales. It could be marketing. It could be legal. All these stakeholders will provide information to the person responsible to make a decision. But the people in the consulted group are not responsible and they're not clearly accountable for making and following through with the decision. Last but not least, I stands for informed. Informed you can imagine is the fire hose which is sending it to anybody and everybody who needs to know but is not directly providing information and is directly not responsible for that decision. This could be partner teams. It could be the marketing teams. It could be anybody in the organization that is interested but does not have direct impact into our accountability into this decision. Here's an example. A good racy would be something like this where the product manager consults with engineering design the data team and the legal team recommends a certain decision and head of product will approve of it. Remember that the head of product because this person is in the A or the approver section the head of product must have the final veto power. If the CEO or the chief risk officer or the CFO has the final veto power. This head of product will now move from A to consult it. And then the person who has the final approval authority will move into A. There can only be one A. In last but not least informed. Informed is just sending it to every other product manager in the organization who wants to know and your general manager or your GM so that they have the information on hand and understand why a certain decision is made. People may seem like a small topic but it's very very important. This speeds up decision making because you can see that the product manager is directly responsible and it's accountable for making that decision. This people section negates some of the problems that I've mentioned in the previous slides around who is making a decision is it the most senior person people not making decision fast enough. People this section right here solves for that where the product manager was responsible the head of product that proves it decision making continues on quickly and efficiently so that the rest of the teams can be unblocked. It's very clear who makes the decision under this section of people. A stands for alternatives. Alternatives is super easy to explain.  A very important aspect of this alternative section is that just the options must be mutually exclusive or collectively exhaustive or Misi. Misi just means that option one is distinct enough from option two and is distinct enough from option three that somebody looking at option two and three will obviously see that they are two separate options and not two of kind of similar but not very different options. Let's look at an example based on our video subscription example. The first option to increase net subscriber growth is to place an ad on top of the video player on the website. We list out pros and cons. This alternative is super easy to build takes only three weeks and protects the user experience. The downsides of this option is that it does not incentivize signups. It decreases the subscription value because if you put the ad on top of the video player it's not blocking anybody from viewing the video. A second option is putting the ad inside the video player. You can very clearly see that alternative one and alternative two are extremely different. Alternative one it has the ad on top of the video player. Alternative two has the ad inside the video player. These are clear, clearly different options. And this option obviously has pros and cons as well. In your example, in your job feel free to use multiple options. Option one, option two, option three, option four. Don't go more than three or four because it makes people very, very confused. Put your top three or four options, lay out the pros and cons and move on. The last part of the spade framework is decide and explain. This is super, super straightforward. You just have to pick one of the options, make a decision and tell people why. In our example, we proceeded with option two to put the ads inside the video player. Although this option requires a little bit more engineering effort. This aligns with our aforementioned goal of increasing the video subscribers to our membership service. You can see here in highlighted in green that we chose a specific option that we laid out. Then we explained why we acknowledge the pros and cons that we laid out in the alternatives table and we decide we acknowledge this but we are choosing this because it aligns with the goal that we set out in our settings, in the setting, in the context section of the spade. Before you communicate this decision out to relevant stakeholders, make sure you clarify and you clean up the document. You get perspectives and divergent opinions from the broad group of people that you're going to consult, your lawyers, your data science team, your engineering, your design team. Make sure that they are following along with the decision. Get feedback. Give them access to this document that lays out the options, the facts, the opinions from everybody else. Make sure that your engineers can see the opinions of the marketing team. Make sure your marketing team can see the opinions and data of the data team. Make sure that this document is fully communicated and transparent and public before it's communicated out for everybody. Just like any decision, there's going to be people disagreeing with whatever you decide. Whether you pick option one or two, there's going to be somebody having a different opinion and that's okay. We don't need consensus. We just need a good decision. And a good decision does not mean consensus. A good decision means you looked at all the facts you considered and you weighed the pros and cons, the trade-offs, and you decided based on decision and the facts that you have on hand at that time, this is the best path forward. I remember an anecdote that I read in a book saying that it is silly to make a decision with 40% of information, but it is too late if you wait until you have 70% or more. So the best decision is to make when you have anywhere between 40 and 70% of the facts. Remember that making a decision does not mean that the outcome needs to be as you expected. We need to decouple making a decision and the final outcome. You can make a good decision with a bad outcome and you can unpack that a little bit more. Why is that the case? Let's say we decide to proceed with option two today. Six months later, we proceed with option two and actually the outcome was it was not the best decision. Our goal of increasing subscribers to the video service actually did not go up. We did not actually goal. That's okay because as of now, we only had this much information. We made a decision. We stuck to it. We commit to it. We execute and if it doesn't work out, that's fine. We'll just learn to iterate for the next time. A bad decision is that we don't make a decision. We get into this series of analysis paralysis. That's a bad decision. For the purposes of this discussion, I want to make sure that everyone decouples decision making and the outcome. There are two different things. Of course, we want to make a good decision to have a good outcome, but that's not always the case and that's okay. Address the disagreements in your document and then communicate it. That's the example I wanted to cover today around spade, but there's a lot of other frameworks out there just like voting, a very democratic vote with your team on making decisions. You can weigh a pros and cons tables, but I particularly like to spade because I use it very often. It's a very, very widely accepted framework in the software technology industry and it's very useful. It's very simple, but it makes it so easy to see all the facts to communicate it, to address disagreements and get into the cadence of making the repetitive decision making muscle in your body. So the three takeaways that I have for you all today is get used to decision making. As a product manager, you'll have to make a lot of decisions that I mentioned earlier. You have to make decisions around product, engineering, design, marketing. Get used to it. If you want to be a good product manager, you'll have to do it. Use frameworks like spade to ensure that you have a good logical structured way of making decisions. You don't have to use frameworks, but the frameworks, when they're embedded into your psyche, it makes it so fast to make decisions that eventually you'll become second nature. The last takeaway is apply judgment. Frameworks just like any type of textbook or acronym or whatever shortcut you use or heuristic you use, it's just a framework.  Blindly following a framework is not the way forward, but it can help you. Decision-making frameworks in general help you scale yourself and your organization. As you go from 5 to 50 to 500 to 5,000 people, decentralizing decision making is the key to making yourself and your organization go faster. For example, let's say you have a startup with five people. You can be involved in every single decision, but when you get to 5,000, there's no way you can do that or you're going to be the bottleneck of that entire company. So use frameworks to train your team to make really good decisions so that you don't have to. Decision frameworks also help decentralize decision-making so that your team is happier. They feel that they're accountable for results and they feel like they own that strategy. For example, if one of your team members is able to decide, they're more likely to follow through because they made that decision. But if you make the decision and you expect your team to follow through, but they don't understand why, they're one step away from making that decision, chances are they're not going to feel like you really understand their perspectives and you're not really investing in their career. So give them the ability to make that decision and let them run with it. And last but not least, decision-making frameworks just provide structure and really ambiguous situations. One of the skills that I see good product managers make is that they apply frameworks and structure not just in large and complex situations but also in very simple decision-making. When there is a very minor problem, I can see good product managers go through a framework so that they consider every aspect. A decision framework like SPADE can help you do that with big and small decisions so that on a daily basis, you get into like a decision-making mode and you come off as someone who is structured and a business and product leader. That covers the webinar that I want to discuss with you today. Again, my name is Vincent Liu. I'm a product manager here at Square. If you want to reach out to me, feel free. This is my LinkedIn profile and I hope you enjoyed that talk.


In the previous class, we discussed about the time domain analysis of structures for specified ground motion and in that we described two numerical integration schemes. One is the Duhamel integral, other is the numerous beta method. Both the integration schemes were cast in recursive form so that if one knows the displacement velocity and acceleration at kth time, then one can find out the displacement velocity and acceleration at k plus 1th time. Provided the force acting at the k plus 1th time is known. In fact, in all cases the load is specified or load is known beforehand and therefore there is no problem. The specific advantage with the Duhamel integral was that for the functions which are not integrable for that one has to carry the integration in the Duhamel integration by a numerical technique. In the recursive formulation that was presented in the last lecture, in that it was assumed that between two time stations or within the interval of time delta t, the load varies linearly or the acceleration is assumed to vary linearly that is the ground acceleration and once we do that then we do not have to perform any numerical integration. One can divide the loading into a constant rectangular loading plus a triangular loading for which standard Duhamel integration can be performed in closed form. So, using the responses for then we formulated a recursive formulation for the Duhamel integral. Next, let us see the state space solution in time domain in which the state space equation as I described before is written in this form that is z dot is equal to a into z plus f s where a is this matrix, f s is the excitation matrix and j is the state vector. Now, if you solve this equation in time then the solution of this equation turns out to be this that is z t can be obtained provided we know the response at time t 0 then this is equal to z t is equal to e into a into t minus t 0 multiplied by z t 0 that is known plus this integration. In this integration one can see that it is a e to the power a matrix is involved here also e to the power minus a into s is involved and f g s is the loading function varying between t and t 0 or within the interval. So, this can be now written that in the form of the discrete variable that is z k plus 1 if that is the quantity that is to be known then it is equal to e into a delta t and then multiplied by z k that z k is known and here this integration is performed from t k to t k plus 1. Now, the integration that which we had shown before that is this integration they can be performed in various form the two forms are given over here. So, that one can get two expressions for z k plus 1 this is the first term which involving z k which is known and it is e to the power a delta t and this is the result of the integration which we were discussing in the previous slide. So, the other result for the integration could be in this form now out of these two forms this form is preferred because this is simpler in the sense that it does not require the inversion of matrix a. Note that the excitation which is to be known is f g k that is the excitation at k at time t now e to the power a delta t is difficult to obtain as such because a is a matrix but e to the power a t can be obtained in this particular fashion that is phi multiplied by e to the power lambda bar t into phi inverse or where phi is the eigen vectors of matrix a and lambda bar is the eigen values of matrix a and e to the power lambda bar t in fact is a diagonal matrix in which each diagonal is of the form of e to the power lambda bar i into t where lambda bar i is a ith eigen vector. So, the size of this matrix is same as the size of matrix a and one can obtain the eigen values and eigen vectors for the matrix a which is known and in this particular way one can obtain the value of e to the power a t. So, in the entire problem the numerical computation which is somewhat intensive is the computation of e to the power a t once we are able to obtain the value of e to the power a t then rest of the formulation is very simple and one can write down the value of z k plus one provided one knows the value of z k. Once you have obtained the values of z at a particular time station that is the state of the system sk and x dot k then we go back to the second order differential equation that is the originals equation and from there we obtain the value of the acceleration. Next let us come to the frequency domain analysis in the frequency domain analysis the analysis is performed for the structure for a specified ground motion and in there the input is the frequency contents of the ground motion and how to obtain the frequency contents of the ground motion with the help of Fourier series and Fourier integral we have discussed before therefore you know how to obtain the frequency contents of the ground motion these frequency contents go as an input for the analysis. The frequency domain analysis in fact provides a steady state solution for the structure it cannot take care of the transient phenomena that is the transient part of the response that comes before because of the initial state of the system and therefore for short duration excitations like earthquake or any other impulse kind of loading which is a short duration loading for that the frequency domain solution may not be applicable that is the steady state solution there does not represent the true response of the structure. The influence of the initial condition that is the transient part of the solution they are plays an important role because for the short duration excitation the interval of time t for which the free oscillation takes place they are the free oscillation may not become very small or may not die down therefore the steady state solution using frequency domain analysis may not be applicable for short duration earthquake. However there are many cases where one can get reasonably good estimate of the root square value of the response and the peak value of the response using the frequency domain steady state solution and that is a condition where the structures period if it turns out to be much smaller than the predominant frequency of excitation then these frequency domain analysis may reasonably provide a good estimate of the RMS and peak responses. So we will generally use the frequency domain analysis for such condition and in most of the practical cases we find that the structures period is smaller compared to the predominant frequency of excitation. Now if we recall instead of Fourier series analysis for finding out the frequency contents of the ground motion we introduced FFT algorithm or the Fourier transform algorithm in which the parent equation is the pair of Fourier integral which is given in equation 3.76 and 3.77 and the discrete part of the Fourier integral is the one which is given in equation 3.78 and 3.79. Once we give the x double dot gr that is the ground acceleration sampled at delta t interval and put it into FFT then the FFT gives us a frequency contents of the ground motion in the complex form and if that frequency content in the complex form is given in I FFT then the I FFT gives back the original time history of ground acceleration. So we use these pair of the discrete Fourier transform to obtain the frequency contents of the ground motion and the solution here is then called a frequency domain FFT solution of the problem. That means here we are using frequency FFT solution here we use two important component of it first one is the use standard result of e to the power i omega t excitation for a single degree freedom system and in the second stage we use FFT and I FFT to get the response of the structure in the form of a time history. I FFT provides the response in the time scale. Now in order to understand the procedure let us look into the standard solution for a to the power e i omega t. So this is known as the complex harmonic function a into e to the power i omega t. So if the single degree of freedom equation is subjected to a complex harmonic function instead of the ordinary harmonic function then the steady state solution part of it can be written as X is equal to X 0 omega into e to the power i omega t. Since the excitation is of the form of e to the power i omega t the response also will be of this form. If we differentiate it once we get X dot if we differentiate it twice we get minus omega square X 0 omega e to the power i omega t substituting the values of X dot and X double dot into this equation we will end up in a equation like this k minus m omega square plus i c omega into X 0 omega is equal to a. Now once we have this equation from this we can write down the value of X 0 omega as h omega multiplied by a where h omega will be equal to the inverse of k minus m omega square plus i c omega and in fact if we make this inversion then it will come in this form as a complex number a plus i b the X t that we are wanting to find out that X t can be obtained now by multiplying h omega with a into e to the power i omega t with the help of this we can get the value of X t. So, the technique is that to get the value of response at time t then we first find out h omega which is known as the complex frequency response function of the system and once it is known then one can obtain the value of the response by this equation. If a happens to be a complex quantity say a is equal to p i omega which will be the case for us for ground motion for specified ground motion then one can write down X i omega to be is equal to h i omega multiplied by p i omega. So, with this background now we will try to explain the frequency domain analysis of the system or the single degree freedom system for a specified ground motion for that first what we will do we will use the FFT for finding the frequency contents of the ground motion the I solved a problem in connection with the Fourier series analysis of the ground motion in order to find out the Fourier spectrum. Let me again repeat it little bit for your recapitulation say if we have got 32 ordinates a sampled ordinates of the ground acceleration then these 32 ordinates are sampled at a value of delta t and is given to the mat lab and we click on FFT. The FFT would provide the 32 numbers of the complex quantity and those out of those 32 complex numbers we choose the first n by 2 plus 1 ordinates that is the 17 ordinates we choose. So, this is your n by 2 plus 1 ordinate over here and the Nyquist frequency or the cutoff frequency is equal to omega n s how to find out the value of omega n s that is n by 2 multiplied by delta omega is equal to the Nyquist frequency where delta omega is equal to 2 pi by t where t is the period of the excitation. So, here the period is taken to be is equal to the entire duration of the excitation plus delta t after that the deputation takes place. So, in other words if t is the duration of excitation then t plus delta t becomes the period and that also happens to be is equal to n into delta t. Now, with this first ordinate first set of ordinates from the FFT we obtain the value of H omega and the for each value of delta omega or each value of omega at an interval of delta omega we obtain the value of H omega and then we multiply this with the frequency contents of the ground motion. The frequency content of the ground motion will be equal to the G omega that is again the first 17 ordinates of the frequency contents of the ground motion that will take at an interval of delta omega and multiply this with H omega. The H omega can be obtained from the equation that I had shown before that is k minus m omega square plus i c omega and inverse of that that gives you the value of H omega. So, once we get these multiplication of this quantity two quantities then this is given as an input to IFFT. In the IFFT what we do is that after the 17th value the next value that we give over here or the next ordinate that you consider over here is the complex conjugate of this one. Similarly, this value ordinate will be the complex conjugate of this. So, these will be the additions that has to be provided from the information that you have got that is the multiplication of these two quantities at different frequencies and we take complex conjugate of them and place them in this particular fashion. So, the last value over here will be equal to a value which will get just after delta omega interval that means this ordinate not this ordinate. The ordinate at zero in fact is the repetition ordinate from where again the entire thing repeats. Therefore, we take up to this. So, in that fashion we can have say if we take 32 sample values of the ground motion then we will get 32 H omega value and capital H omega value. This capital H omega value is nothing but the small H omega value multiplied by the frequency contents of the ground motion and this multiplication of them is a quantity which is a complex number. Now, once we give this 32 complex number that means on this side we have complex conjugate numbers then the IFFT would give us the response of the single degree of freedom system to the specified ground excitation of XGT, XGT is the ground motion. So, that is the frequency domain analysis and in that frequency domain analysis you see that we are using the FFT algorithm and also we are using the standard solution which is known for a single degree of freedom system excited by e to the power i omega t. So, the frequency domain analysis can be carried out in a very routine fashion using FFT and it requires very less time compared to the time integration techniques. So, the cases where we are able to use a frequency domain analysis we go for that because these days the FFT algorithm is available in most of the standard libraries in the computer. The summary of the frequency domain analysis is given over here that is sample X double dot GT at an interval of delta t that will give a value n values of the ordinates then input these sample values in FFT then consider the first n by 2 plus 1 values of the output and we get the Nyquist frequency and the Nyquist frequency is given by this and we obtain H j i omega which is the complex frequency response function for the single degree of freedom system which is nothing but is equal to k minus m omega square plus i c omega inverse of that then obtain this quantity that is X j i omega is equal to H j i omega multiplied by X double dot G j i omega X double dot G j i omega is the on the frequency contents of the ground motion then add appropriately the complex conjugates as I explained before then put it as an input to I FFT and I FFT would give the values of the response in time. Since both frequency and time domain solution can be used for the solution it is expected that there should be a relationship between H omega and H tau. H tau specifically arises in the Duhamel integration that is for the unit impulse the integration that we perform for obtaining the Duhamel integration is called H tau and there exist a relationship between the frequency response function and H tau and this relationship is nothing but the again the Fourier pair of Fourier integral only difference here is that 1 by 2 pi is not associated with the first equation which is usually which is the generally the case for finding the frequency contents of any time history but here 1 by 2 pi is associated with the I FFT part of the equation. So, FFT part of the equation and does not have 1 by 2 pi whereas the I FFT part contains 1 by 2 pi. However, if we interchange this then also the results of the Fourier first Fourier transform do not change. Let us look into a response of a single bay portal frame and in this single bay portal frame we are interested in finding out the response of at the the sway response of the frame that is you subjected to a ground acceleration of x double dot g the two masses are lumped at these two points and the EI and L values are specified for the inclined leg portal frame the natural frequency of the system is equal to 12.24 radian per second and the damped natural frequency is 12.23 they are very close to each other because the damping is coefficient is very small and the initial condition for them is given as x 0 is equal to 0 and x dot is 0 is equal to 0 that is the initial condition is given and the excitation is sampled at a time interval of 0.02 second. Now, first let us see how we can obtain the natural frequency for such a system which is an inclined portal frame. So, the stiffness first let us take the stiffness of this frame to corresponding to the sway degree of freedom. So, that will be the key value and they are a combination of these two masses will form the effective mass and root over k by m is equal to the value of the frequency natural frequency. In order to obtain the stiffness corresponding to this degree of freedom let us try to recapitulate how we can find out the stiffness of the system. We are interested in obtaining the force required to produce unit horizontal displacement over here. So, for that what we do we make this instantaneous center of rotation over here and give consider this as a set square and rotate this about the instantaneous center such that the horizontal component of this movement becomes equal to unity. Now, this movement will be perpendicular to this member or perpendicular to this dotted line. This movement also will be perpendicular to this member or perpendicular to this member. In that case the length of this and this they remain unchanged length of this and this remain unchanged according to the small displacement theory and in this triangle if these two lengths remain unchanged after the rotation then the third length that is the joining this point and this point third length also will not change. So, if they do not change then the length of this member remains unchanged length of this member remains unchanged because this point is moving perpendicular to it. So, therefore it remains unchanged. So, all the three members after this given rotation about the instantaneous center will not undergo any change of in length therefore the condition of in extensibility is maintained. Now, once we have that then one can find out the forces that is generated at these points due to this rotation. So, we have first let us consider this beam in this beam the relative displacement in the vertical direction between these two points can be worked out from the knowledge of this and this and for that relative displacement we will have a force or shear force acting in this direction over here and shear force acting in this direction over here and there will be a rotation also because of this rotation at these two points there will be again a shear force which will be acting in this direction and in this direction. So, in the beam finally we will have a shear force acting in this way and acting in this way. Next let us take these particular inclined leg in the inclined leg since it is moving perpendicular to the member over here there will be a moment produced and there will be a shear force in this direction plus there is a rotation rotation say is in this direction then this rotation will give rise to a shear force in this direction. In the same way one can find out the shear forces due to the displacement and the rotation at this point for this inclined leg. So, they are these forces. So, we have got now three forces acting at this joint and three forces acting at this joint. These three forces now are resolved one along the this leg that is along this direction and other in the horizontal direction or the direction in which the sway takes place. Similarly here also these three forces can be resolved one along this length and other in the horizontal direction that is the direction of the sway. Some of these two in fact will be the force which will be required to produce unit displacement and that defines the stiffness of this inclined leg portal frame corresponding to a sway movement of unity and these forces which are coming on to the leg they simply pass to the support and there is no shortening of the member in this direction because it is assumed that the members are inextensible. So, in this particular fashion one can find out the value of the stiffness for the inclined leg portal frame. Now once we get the stiffness for the portal frame then we have to find out the effective M. Now effective M mind you is not is equal to M plus M because if we give a unit displacement a sway unit displacement in this direction then the entire system would move like this. This point will come over here and this point will come over there as I had shown before because the inextensibility condition has to be satisfied and for that the inertia force which will be generated will be in this direction minus M or M multiplied by delta double dot where delta double dot is the acceleration caused due to the unit displacement in this direction. Similarly M multiplied by delta double dot will be the acceleration produced in this direction and hence mass multiplied by delta double dot will be the inertia force. So, these are the two inertia forces that will be generated over here because of the movement of the structure in the horizontal direction maintaining the inextensibility condition and these two forces are resolved one along this direction and the other along the member similarly here it is resolved in this direction and one along this member. So, some of these two inertia forces are the mass matrix or the mass equivalent mass of the system. So, the omega n for calculating omega n we make use of square root of k effective divided by M effective where M effective is this one and the k effective we have found out before and then one can convert the entire system as a single degree of freedom system with a spring having k effective and the mass which is M effective and the dashpot can have a constant which will be equal to twice xi M effective into omega n whereas xi is the specified value of 0.05. So, we now solve this equation by various methods that I have outlined previously. So, if we use the Duhamel integral then in the Duhamel integral the type of recursive formulation that we used or proved is this is the Duhamel integration if you for the capitulation we use the q k plus 1 is equal to f n multiplied by q k and h n multiplied by f k plus 1 f k plus 1 is known that is for the numerous beta method and for the Duhamel integral we have A multiplied by q k. So, this A matrix is the matrix which is given here is the solution this is the A matrix this is the A matrix for Duhamel integration and the f n matrix is the numerous beta method which is given here and h n is this matrix. So, with the help of these matrices one can find out the response at different time steps in the state space solution we have seen that we require e to the power A delta t and A is this matrix state space matrix the Eigen vectors for this matrix is given by phi and is obtained like this this is the Eigen vector and the Eigen values matrix multiplied by delta t that is that we can e to the power A delta t when can be is obtained like this. So, with the help of these quantities one can easily obtain the response of by in the state space solution at different time t. Now, the time histories of the responses that are obtained is shown over here I think the f f t solution also was carried out that I have not explained over here, but explained before how to obtain the response of the system using frequency domain f f t analysis and then it was compared with numerous beta method and we found that for both the cases the results were exactly the same. The reason was that the initial condition that was taken for this problem was that it was at rest that is there was no displacement and no velocity at time t is equal to 0. Therefore, the both the frequency domain solution and the numerous beta method solution they provided nearly the same result. Next we come to the response for the multi degree of freedom system and in the multi degree of freedom system as I told you before we classify the problem into either a single point excitation or a multi point excitation. In the single point excitation we have i influence vector or i influence matrix that generally is consisting of 1 0 1 0 etcetera. For multi support excitation the i is replaced by r, r is a matrix that is called the influence coefficient matrix and that is generally obtained by performing a separate static analysis for the differential support movement at the base and from that how we obtain the value of r that we have explained with the help of a few example also. Now in response analysis for MDF system again we can have two types of solution one is a direct solution other is a modal analysis. So first we will take up the direct analysis in which the damping matrix must be explicitly known because the equation of motion becomes mx double dot plus cx dot plus kx is equal to minus mi or mr into x double dot g. So the c matrix must be known and c matrix is obtained by assuming it to be a Rayleigh damping that all of you know and that is equal to alpha times the mass plus beta times k. And this alpha and beta values can be obtained from the two frequencies of the system it depends which two frequencies you take most of the people take the first two frequencies but it is not necessarily that one should take first two frequencies. In fact any two frequencies can be taken and which two frequencies we will describe the damping matrix best that is a matter of debate and research. But generally for you for the solution of the problems we satisfy ourselves by considering only the first two frequencies. For modal analysis we will require more shapes and frequencies that we will discuss later. Again in the case of multi degree of freedom system one can have a time domain analysis as and frequency domain analysis. So first let us discuss about the direct analysis in time domain. In the direct analysis in time domain we have the this is the equation that we are trying to now solve. Here the R is the influence coefficient matrix for multi support excitation. If we change R to I then it will become a single support excitation system and this is discussed over here say if it is a single point excitation system then minus mi x double dot g that is the right hand side force vector and if we expand it this is a diagonal mass matrix I is a vector for the multi story buildings 1111 x double dot g that gives a right hand side load vector as minus m1 m2 m3 up to mn into x double dot g. So we have got n number of forces and these n number of forces is such that its components each component of this force vector is m1 x double dot g m2 x double dot g m3 x double dot g so on. And if one has to perform the solution in frequency domain then each one of these forces has to be Fourier synthesized or effective algorithm is to be used for each one of these forces. If we consider the multi support excitation then the this is the form of the equation that is minus m into R into x double dot g this is the mass matrix diagonal mass matrix the R matrix say we are considering 4 support excitation then it will be a n multiplied by 4 matrix n is the number of degrees of freedom and the R matrix would look like this and x double dot g1 x double dot g2 and x double dot g3 and x double dot g4 they are the support excitation. Once we multiply this vector with this matrix the and then multiply with the mass matrix then we find out a vector of size n and each element of the vector we will have terms like this that is m1 multiplied by R11 x double dot g1 R11 into x double dot g1 plus R12 into x double dot g2 R13 into x double dot g3 so on. So that is the first element similarly one can obtain the second element third element so on. So these quantities R11 m11 they are all known therefore the this particular vector is known. So one can construct n number of time histories and this n number of time histories would be the excitation vector now if we again go for a frequency domain analysis then this n number of time histories are to be Fourier synthesized or FFT algorithm is to be used for finding out the frequency contents of all these n time histories. Two cardinal equations that we use for numerous beta method are given in equations 3.86 and 3.87 the equations are written using linear variation of acceleration within the time interval delta t. Note that the variables of the equations are vectors using these two equations the numerous beta method can be finally cast into the recursive form of equation as we had done for the case of single degree of freedom system and is shown in equation 3.92 the difference between the two cases that is equation 3.92 and the previous equation for single degree of freedom system is that here the FN matrix contains terms which are all matrices. For example S is a matrix then G is a matrix S is equal to G inverse K, Q is equal to G inverse C and you can see that the Q matrix is appearing in FN matrix. Then there is a T matrix which also appears in FN matrix and T matrix is given by G inverse M into R where R is the influence coefficient matrix. HN matrix is given by equation 3.94 that is minus T into delta T square by 4 then minus T into delta T by 2 and minus T. The G in this matrix which requires an inversion to find out S, Q and T this G matrix is simply is equal to mass matrix plus delta into C matrix into delta T plus beta K into delta T square. Now with these matrices defined one can find out the value of the responses at K plus 1 th time station given the value of the responses at the K th time station and the direct time domain analysis using numerous beta method can thus be obtained. Now let us come to the frequency domain analysis using FFT. In the frequency domain analysis the right hand load vector for single support and multi support excitations are given in equation 3.95 and 3.96. In equation 3.95 the term which are shown are very straight forward with x double dot G as a single time history whereas in equation 3.96 the equation is little involved in the sense that the terms of the load vector turns out to be the mass of a particular force say first force multiplied by the sum of R11 x double dot G1 plus R12 x double dot G2 and so on where R11, R12 etcetera they are the of the R matrix that we have obtained before. And the excitations are x double dot G1, x double dot G2 that is the excitations at different supports. Similarly one can obtain the second element of the vector that is the mass of the second floor multiplied by similarly R11, x double dot G1 etcetera. Equation 3.97 gives the necessary equation to find out the frequency content of the displacement vector x i omega and that is equal to h j i omega multiplied by P g i omega where P g is given by either equation 3.95 or equation 3.96 depending upon whether the load vector is defined for a single point excitation or a multipoint excitation. Note that h for computing h j i omega one has to obtain a complex inversion of a matrix given by equation 3.98. Using the steps as we have discussed before for single degree of freedom system x t then can be obtained using f f t and i f f t. We use this relationship that is x j i omega that is the responses frequency components of the responses is related to the h j matrix and the P g i omega vector h j i omega matrix is in fact the matrix which will be k matrix minus m omega square plus i c omega which you have got for the same kind of expression you have got for the single degree freedom system only replace k m and c by k matrix m matrix and c matrix and inverting this matrix you get the value of h j omega and this is a n by n matrix if the number of degrees of freedom are n. P g i omega will be the frequency contents of the ground motion and this frequency contents of the ground motion will be the different time histories of the ground motion or the forces that we get under the right hand side. So, those different forces all of them are Fourier synthesized and we get the value of P g i omega and this again becomes a vector for a specified value of omega this becomes a matrix for a specified value of omega if we multiply these two then there is it becomes a vector and for a specified value of this omega one can have a vector of size n. Then each one of them can be taken out separately and the first n by 2 plus 1 quantities of x can be written down or put in FFT algorithm and then we add on to that the complex conjugates of them as I described before. Then we have the complex set of x j i omega for the response one and put it into I FFT the I FFT gives the response of x 1 in time that is a we get a time history. Similarly, we take x 2 and again add on to it the complex conjugates give it to I FFT and from there we get the time history of the response x 2 t. So, that way one can get the value of the responses for all the n degrees of freedom. So, that is how one can obtain the response of a multi degree freedom system using FFT and the frequency response function matrix over here.