Great. We're going to call the, it's like we're getting to order and for stuff that's public comment, just anything on the agenda. I have. Okay, go ahead. Just two, I guess. Comments one, I'm hoping that the town report will have the different capital reserve funds listed like it has in the past, and any plans uses of those funds. I think it did two years ago I'm pretty sure it did not have that information last year or so. And I've mentioned this before in another select board meeting, but just, I'm strongly urging the town to include pavement maintenance, like crack ceiling as part of our regular work every year, because, like I've observed some fairly new pavement like two or two years old that already has like a crack down the middle, and basically if we don't take care of things like that, the water gets in there, and we'll end up, we don't spend a little bit of money like every year on that type of work, then we end up spending a lot more money down the road. So, just urging the town to do that kind of preventive maintenance, the road projects. I don't have any more public comment. Anybody online with public comment. They're all staying muted. Good. Approval of the agenda. The approval of the agenda. Second, all those in favor. Mr. Terry's consent calendar meeting minutes Lister's office, no appeal or certificate and warrants. The consent calendar. I will second. All those in favor. Business. First up in the update on the town police department. Nobody has noticed we did not have Trevor with us tonight. He is out of the COVID and upset that his kids will share COVID with him but not their Halloween. Just given a status of where we're at and what we're up against, starting shortly after November. We got a lot of folks reaching out with fears of what was going to happen with the Orange County Sheriff's Department. We've monitored it for a while and noticed that some of the things that were being talked about were happening. So we did get a little bit more than just monitoring and started doing some looking into met with some other departments that surround us met with the state police. Bottom line, nobody could provide the service. Everybody was tapped out. Most of them are short staffed. And the state police especially. So, there was a little bit of effort but not a lot put into what are other options can we bring our department back to life and what does that look like. It didn't take long before we heard that there was not enough. There was a lot of employment by the Orange County Sheriff to even do the basics that they have to do at the county level, saying nothing about meeting their contracts. And there was a lot of discussion around that they were going to let the contracts go, but we wouldn't know until the sheriff was on board. We did fair amount of scrambling and short period of time, that's safe to say. To look at what we do and how we stand back up the police department that the town had for quite a few years. I did look back in the last budget that we approved in the town with the July 18 to June of 19 budget for the entire department. And then we went to the contracted services through Orange County, which was a cost savings, obviously to that district. But now we find ourselves in the boat of having to stand up some level of law enforcement to have that service available in the town. So, we do have some staff that have been hired and Scott can probably fill in some of those more than me because I didn't get a chance to talk to Trevor about it. But we have hired a spot on as the chief. And we've hired the admin person. At this point, but no other officers. And some of what we're happening, what's happened to happen is to reactivate everything that got deactivated when we got rid of the town police force. So all our ability to access all the different computer systems to get different information, even to be able to do things like buy the firearms. Do those type of functions to get it. And, you know, Scott was telling me today we came to apply for some of these grants that will help her pay for these things, because that's number hasn't been reactivated yet. So there's a site visit coming up on Tuesday that will get us one step closer to that. And try to check some more boxes and hopefully get it. This is an agency or I remember. So is that a state certification. Well, it's kind of the update. But the state is one of the gatekeepers that we get through. And they've all been great. They're all helping like this is this is pretty quick that all this is finding its way through the process but just getting over those hurdles I think I want to speak for you if you want to tell them or you're welcome to but you know we believe it's in the three to four week range that will be happening. That's on the very, very, very, very fast track. A lot depends on what happens with the site visit. If there's maintenance because we have to secure the building with the newer renovations and things that will be accountable for the old stations grandfathered in back years and years ago. I think that we're reinstating this we happen to do all that stuff and I don't know what the extent of that is until the second is what we're looking at and that's Tuesday if they're in the morning. What might be some examples of that. I think there are like have to make sure that there's no access a lot of pasta for your mess for the proper signage internal camera inside the building overlooking all exit points and entrance points. We have to make sure like the windows are secured. Those kind of bits and pieces. We can't have any screens that are facing windows. No cameras that look at when our best greens it's all information privacy stuff that's kind of going on right now. The goals of the CDS policies need to be on a book and everybody's up to speed and being able to access that information. Even as far as like janitorial stuff coming in to ensure they're all better through the proper systems that they can access the building without export, or get probably printed and vetted through the stages programs to be able to have access to the building. Very similar to like moving up a fire station. It was perfectly fine where we were but it's going to something happen that changes your scenario now you've got this new standard. Is that pretty let me just make sure we're through with the party. That's where that or I is standing that's our, our major hangout without that or I as tree said we can apply for law enforcement based grants. I can apply for department issued weapons. So like the bulletproof best program that we just tried to handle people apply for and do it without that or I, without that or I number were kind of dead in the water. So we got it we're trying to push that out as fast as we can. Any questions on that. Kelly. I just want to say thank you all for being on the select board and handling all of our town's business. For all of us who feel too busy to to run I do I do deeply appreciate that a lot. I want to say I live in the police district and I pay the tax bill and I'm pretty concerned about a $775,000 police department for the police desperate. I was on the committee in 2018 19 that disbanded the police department. We met for months. We had three public hearings about what our needs for policing were. And we learned quite quite a few things some of them may be relevant information to you all, which I'm hoping Larry can remember some of it. So this budget as as I recall, there are 1500 people living in the police district is that, do you know, is it 2500 people, is it 1500 taxable properties. Yeah, I'm going to ask the listeners to let me know that information but it's a very small group that has to bear this this burden. And when we had these public hearings, a lot of people there are a number of members of the committee from outside the district which I was like do you pay a dime for this why are you weighing in, but whatever like we did have those folks here and a lot of people from outside the district came and testified for us, and it was very very crystal clear one takeaway I remember clearly is that people outside the district don't want to pay a dime for policing. So, they're satisfied with relying on state police for services so the one item in the budget that you have is a $100,000 general fund transfer that that basically shifts the cost to the whole town. So that's the budget, and I'm not sure that those that people outside the district. I know for a fact that they don't want and that that was the status of what we took away from that meeting that they don't want to contribute to policing they don't want that tax burden for the report for them. So that's, that's one concern I have and then of course I'm, I, I don't think that the village, I'm calling the village police district requires for full time police officers and an administrator. The district disbanded the department, just as the department was asking us to float a million dollar bond, asking the district voters only to float a million dollar bond for a new building, when, like, as far as I can tell, but you all know better than I do, we've got like some problems with our water system, and we have a lot of like seems like infrastructure projects we have to be working on so that that was like a complete concern to me was like, where are we going to get a million dollars among these, I think 1500 taxable properties for a building to do this so this budget of 775 is really high out of the gate, and it's only going to grow. And I expect that someone eventually will be looking for a building. And again, it's, it's, it's, it's a really small number of people who have to bear the costs of this. So that's, that's why I will be voting no to this expenditure come come town meeting day. We do a lot of work for months to hear from people, and the types of things people want the types of police services that people wanted. And the types of calls that our police officers were getting would be satisfied by a constable, whether it's a constable who is a lot of licensed, or a certified law enforcement officer like Scott. And our size, our police district size 1500 people I think in Brookfield have two constables and they need all those needs which can be service of process, animal complaints both wild animals and domestic animal constables, neighborhood disputes, they do welfare checks. These were all the types of things that people who came in, testified that they were worried if we didn't have a police department, those are the services and they would be missing. And constables under Vermont law can also be law enforcement officers and have those have those law enforcement powers as well. So, if this budget reflected to police officers. I'd support it, but this is way too much of a burden for the for the district voters. And it sounds like it's a dungeon. I'm telling you all this but I thought you should think about that or know know that anyway. To help you understand a little bit about why there's more money put on the other side. Orange County had a contract to provide services out there. It was $25,000 a year to provide services to the rest of the town, but we got a lot more than that. So, your officers that were here in town, are driving through the rest of town, every day to get back to Chelsea or to come back over to go to the courthouse or whatever they took different paths to cover more areas. So we got a lot more of that service we're not getting any service from Orange County now. So, 25,000 isn't even going to be enough to help us in what's going on out on the road part of our town. There's at least two big drug houses outside of the town on the other side of the interstate. There's one outside of town here in the village that's just south of the line that require a lot of police service. Our school requires a lot of police service. Our hospital requires a lot of police service. This town's service demand is much more than just navigating neighbor complaints and those type of things. And there's other ways you can resolve them. You're completely right. You don't need a SWAT team in there to deal with those things. But the level of calls and the type of things we're seeing in this community right now, going without law service is not an option. And I do appreciate that, but it's patently unfair to require the people living in the paying taxes in the police district to provide police services to this extent to a school that three towns use. Yeah, so what you see there in the budget, it has some money in there allocated to the other users, but we don't have any way of knowing that. Yeah, literally this thing has been a whirlwind to put together. And I appreciate that. And so your numbers are probably a little high because we have no idea on some of this stuff. What we did was went back to the old budget. If you look at the last budget we approved for an entire police service. The, the to be raised by taxes was 516,000. Yeah, we're proposing I think 499. So what we, we have meetings coming up with the school, we have meetings coming up with Gifford and Claire Martin Center. Those are some of our power users if you want to call them. Yeah, I understand. And so there's a conversation that's already been started with the hat outreach with them. We've started that kind of, you know, is it's not fair that we're paying. And so they're kind of receptive kind of, you know, yeah, oh, our budget and we're already in trouble. And just, just one last thought is that, and just to alert you and I can't quite remember the details but my, my recollection and again Larry, Larry was there is that there is some kind of perhaps binding legal situation where that has to do with whether people outside the district. So you know what I'm talking about at all. Is this ringing a bell. Well, I felt like the people outside the district could not be taxed, unless they voted to be involved but anyway, I mean, I'm sorry to be like in the current budget. We have a line item of $100,000 from the general fund for that goes into the district police like that. Right. And that, I mean, the voters will vote on it. I mean, if they don't like it enough that they want to go down the budget. I mean, they could. Yeah, that's their recourse. Yes. But that's basic but it was basically otherwise it's sorry, you know, it's a budget support decision to create the budget the way we did. Yep. It's there in the Emory was looking at some of that for us on the on the taxing, because it's there. It can be done two ways. They can contract with the rest of the town to provide the service also. So there's two ways to get access to that money it could just become part of the budget, and it flows into their budget. If we're able, if not we contract to provide the service to the rest of the town. The problem is there's nobody else that can provide it. So if we're going to go through all this effort and we're going to do all this and we're going to use our money to support some of this, the, it needs to function. And we can't say the rest of the town, sorry, but we have this department stood up here and people but I can't cross that line to help you out knowing the state police can't get to you and nobody else can. So we're trying to, we do know there's a nuance of which way you do it but it's in the budget that way. So we have the ability to implement it whichever way we can. Thank you. Training training. Hang on, I got one more in the room first. Yeah, I guess I share the same question about the sort of the scale of the department, you know, or I believe it's what for officers, plus an administrative person. I have sort of that same question. I have a few sort of specific questions about the budget and the way that it's warned on the town meeting warning for, you know, for FY 24, because, you know, you know, 700, it's 771,387 of which 271 900 shall be from non, excuse me, non tax revenue. So, and part of that 271 900 is the transfer from the general fund. So how is that non tax revenue. It's not the police department. It's not the police district. Okay, so I thought you'd say that. Yeah. So my question is so I live in police district as well. So I'm going to pay my general fund taxes. And I'm going to pay my police district taxes. And part of my general fund taxes are going to go back. So I'm going to pay kind of double in a way. Right. I guess, I guess I disagree with the characterization that $100,000 is from non tax revenue. No, you're, you're right. The grand list for the police district is about a third of the size of the total grand list. So when we say we're $100,000 coming from the general fund, the people in the police district will pay about on that, you know, overall will pay about a third of that. So, so, so it was the warning like inaccurate. No, no, it's not inaccurate. It's just that you're, you're paying it through your town like municipal bill, not the police district bill. The question is a little bit different than that Larry. I think that's a little bit different than the general fund taxes. But that question is the one that comes on that separate paper to only the police district. And it's about the tax being assessed only to the police district. So the tax being assessed to the police district will raise the 499. And the other revenue is not part of that 499. I think it's a little bit different than the general fund tax. I think it's a little bit different than the general fund tax. I think it's a little bit different than the general fund tax. I think it's in your general fund budget, which you also vote on. I understand that. And that comes from the general fund tax. I think that that is a subtlety that most people will not understand. Unless like the newspaper prints something about it or it's like where does now reach and I think it's a little bit misleading. I'm not. I'm not going to. I'm not going to say that. I don't want to say that. That's going to be a little bit different from the general fund tax. I'm not going to say that. I'm not going to say that. I'm not going to say that. I'm not going to say that. I'm not going to say that. That seems like a little bit different than the general fund tax. The budget. The 50,000 from the high utilization organization payments. That seems like very, very speculative. And then the 100,000 in ARPA funds. I presume that is just going to be in FY24. And in future years that hundred thousands going to then come from 100,000 higher at minimum to what the police district people are going to have to raise or approve. And generally, I mean, I see these types of things going up in the future, right? Salaries will go up, health insurance is going to go up, and that's the bulk of what the costs are. So I just really question, I guess a question I have for the board is, did you consider giving it a town-wide item that's part of the general fund that everyone pays for if you're and that way it's transparent? Well, we talked about it, but we didn't go there. The way you're going to hear from the other side is you guys get all the service. Why are we paying for all your service? And we get that other side of it too, so and the taxing authority isn't there for the rest of the town. I mean, it just seems for a town our size to have almost a sixth of our town budget going to a police department seems out of scale with what, I mean, you cited a couple of things. I would be interested to know about the data behind, you know, what is the police department actually doing? What are the calls? And how much is it, how much are four full-time officers really needed? I know because it and, you know, just one last thing I would say is I feel like, you know, this meeting is the first time that this has really been discussed in kind of an open way. Like, I don't know how people around town, you know, feel about this and in the district. But anyway, I think there needs to be more information out there about sort of why this decision gets made and maybe people will buy into it and maybe they won't. I agree with you. It needs to get out there and more information, but just understand like you're talking. We were, a lot of this conversation was about contracts, about like, what do we do? How do we do it? It was a lot of stuff that wasn't able to go out to the public because we were also balancing the, you know, the sheriff, the new sheriff of elect who wouldn't work with us or talk with us. Right. What's he planning? Is he, does he suddenly have this policy that's going to ride in and populate all his position to be able to do the service? There were so many unknowns and we're sitting here planning multiple scenarios waiting and actually the resignation letter only just came through. So it's, I guess, yeah, it wasn't a good scenario for us either. Yeah, I appreciate that that you're in kind of a tough and with the timing of getting the town meeting warning done and is there any like repercussions for the sheriff breaking the contract? It's required to give a 30 days notice, operated a day and a half with nobody, no notice, no nothing, and then send us a letter. I mean, what are we going to go after? Like, I know it's a tough one. I was just going to say that we did feel like we needed to act pretty quickly and if there could be rapid action, it's still going to be a while before we have any police services again in town. And so I think we did what we thought we could do under the time constraints. And I'm doing this as really the beginning of the conversation of what police services look like here in Randolph and who pays for it. I don't see why what we have right now needs to be what we do forever going into the future. I would like to see some adjustments made in the future. And part of that will depend upon what our experiences over the next several months as we start to then construct a plan for the following fiscal year. It's very hard to scale back as opposed to scale up. So if you serve with five municipal employees out of the gate, it's very hard to be like, oh, actually, we just need to. Well, we're probably not going to go down because I found that with with three officers from Orange County that that provided a good level of service. And the reason why and please tell me if I'm not getting this straight, but because I have the same question about why don't we just have three officers or maybe Scott could actually answer this question better than than I but I'll give it a shot to practice. That with the Orange County Sheriff with with providing us three officers was able to keep three officers on duty, you know, every week year around. But because we don't have other staff to draw upon if we have three officers on duty all the time. If one of them is sick, if one of them was on vacation, if one of them is, you know, out for whatever reason or if we need another officer, if there's some sort of an event or something like that, we have no one to draw upon. So we want to maintain except our state police is is is is is down quite a bit. We can't they told us we can't rely on them basically for anything. So so the idea is that four officers basically gives us essentially the coverage which we've been used to having. And then the administrative person is that was also a service which was provided by the sheriff's department. And and if we surround the police department, we need that person. And and I don't know if if this has been figured out in any detail, but I know there was we've also talked about the possibility of that person being able to fill some other holes in our overall administrative deficit in the town office. So that person would not necessarily be a full time, just police department person. So those wages would wages when they were working on other stuff wouldn't be charged of this budget. So that's the background in terms of, you know, how we came to the decision we made. And I mean, I'm open to, you know, looking at staffing levels in the future, but given what we knew when we were making these decisions, that's that's what went into that. And in terms of what the size of the police district is and who pays, I'd also like to see that conversation continue into into the future. Marius. Training. Hello. Yeah, go ahead, Marius. Oh, thank you. I had a couple of comments. First, I appreciate the direction that the select board is taking here. And I know you're in a difficult position. And I think what you're doing is right. And I don't think you have to solve every problem in the in the next two days. But you do have you do need to get a police force in place in the next few days. And I think you're you're heading in the right direction. There's some lessons to be drawn from this, I think. One is that it's never a good idea to turn over police services in a community to an elected public official who's elected by people outside of that community. And it should be obvious why that's a problem now. But secondly, when it comes to the extent of the need, I think we need to take a look. I've I've looked at the contract pretty closely between the Orange County Sheriff's Department and the town. And it looks to me like the hundred and twenty hours is for I would I would almost describe it as casual police services. If there's any documented incident and the term is not defined in the contract, that's billed on top. And is a documental incident is what is that and how many of them have there been and how much have we actually paid for these Sheriff's Department police services? And what does that tell us about what we actually need in terms of police protection? I don't trust the argument that that we we're a safe community who doesn't need any protection. We may be a safe community, but we need protection. And I strongly support the direction that the select board is going in. I know you may want to make corrections down the line. I'm not worried about over budgeting right now. And as far as asking people outside of the police district to pay, you know, free riders are all the same. They don't want to pay for anything for a service they don't need until they need it and then they don't want to pay for it. So we need to take into account the fact that people who are in the police district are carrying the burden and they're carrying the burden for some of the people outside the police district because policemen respond to calls from outside the district. So let's get this straight. You're going right in the right direction. I appreciate the direction you're going in. I encourage you to continue it. Thank you. Thanks, Marie. I think the other thing we learned is a lot of our policing services are actually being delivered in Braintree. So we may need to have the conversation with them, too. Do we know, as I mean, this is going to flow from the conversation that we've been scheduled, but do we know or Scott can maybe you estimate the degree to which the school and the effort of being within the district as a proportion of your overall activity? We ran the numbers and I apologize. I do not have those in the front of me. If it was to what we did in 2021 and 2022, the average between 16 and 17 calls, I believe, or given alone, the school was more. I apologize. I do not have that number for it. But that range from medical emergencies, mental health or simple control for Veggie Van Gogh or Last Mile or things of that nature as well. School, anything from any reading books to kids, leave programs or unruly kids, incidences that happen, fights or anything else that kind of transpired within those those areas we handled. This is a, well, we talk about it and it's a concern of mine that we have a major regional medical facility within the context, of course, of a rural environment. And we have a high school that serves three communities and 1,500 people in the village and afraid for all of that. I think it's time to make things a little more equitable. I have the conversation much more openly than that. My understanding is the school ready to come to the table and have that conversation. If we're willing to have a conversation about what they would like from us. I mean, the reality of the school is that if they transfer the cost to the school budget, we're still going to be paying for it. We'll double the other times. But so will the other time. Well, those are certainly those are certainly legitimate issues without question. But would anybody in Randolph not want the high school or the hospital to be here just because we have police services? I don't think so. That's not what I'm suggesting. I know I know it isn't what you're suggesting. I'm just saying that the argument that they're here and they're a burden to the taxpayer who has to pay for police services. It's not a really strong argument from my point of view, because we want those services here. We don't want them somewhere else. Mm hmm. So we do. Every time they come in for an application for development, we have to sign off that we have the capacity to provide those services. And I think it's questionable whether we could sign that today for anybody. I think you're right. Mm hmm. And that's why I'm glad you're going the direction you're going. Yeah, it's a it's a tough one. No doubt about it. Any more questions or comments? On this topic, there's nothing for us to act on, but we will be setting a special meeting to go over all the policies and try to get them reviewed and adopted. So the policies on the books are for the town of Randolph Police Department when it went non-exist. So they haven't been updated. They aren't the current loans. They're not the latest and greatest language and whatever. So Scott has been working on downloading all the ones that we're supposed to have and changing them over to Randolph. And they have to be adopted before we've been adopted. So I'm going to end up playing. We'll hopefully get through those, get them read through and ready. Get them out to everybody. There's a lot of them, unfortunately. And then they can set up the time to go through them. We might be able to go back and forth with questions or whatever to try to iron them out better before we need. So any more on this one? Any questions or comments? Mary? Well, thanks, everybody. We move the agenda forward to the setting with Mary Richter, the Economic Development Council. Mary is online. Yep. Hi, how are you all doing? So the reason I'm coming to the Select Board is that the Economic Development Council has not met for, it's going to be about a year. We had done a lot of things during COVID, but initiatives have somewhat fizzled. So I'm just going to go back with a little bit of history. I don't know all of the history of the Economic Development Council. However, I know that I was involved with the R3 initiatives in which we had brought to the town to hire an economic development director. So that happened, and of course, Josh was our first. And then Josh left and Mark Resolbo is now the economic director. After the hiring, it was unclear of the roles between the Economic Development Director and the Economic Development Council. So what I had done was there was no clear direction of who's going to do what. So I did some research and found that there are no towns within Vermont that have an economic director as well as a council. It's either one or the other, but not both. So throughout the summer and fall, I've had discussions with Mark to how we can restructure the Economic Council. And the takeaway was for me to look at the bylaws. And so I had a small core group where we had a discussion in November. And what we found is one, the bylaws are outdated. I think the last time they were updated was in 2015. And then it did not align with the September 2019 town plan. So among the small core group we discussed and the decision was to disband the Economic Development Council. And I had a conversation with both Mark and Trevor. And the goal was to have that be under the Economic Development Director where he would dovetail efforts and initiatives with the local nonprofits, as well as the Green Mountain Economic Development Corp. And then out of that, whatever initiatives, it would be up to the Economic Development Director to create like an Economic Development Advisory Group so that it could consist of businesses and community outreach and that there could be core items and tasks to go forward and to be more productive than what we have been having. In addition, when I've held meetings I've never been able to have a quorum so business could never be conducted. So that is why I'm coming before the Select Board for consideration to disband the Economic Development Council and move it under the direction of the Economic Development Director who then would then develop an advisory group for different initiatives. And thank you. And I'll answer any questions that you may have. Any questions on that? I can confirm what Mary just said because of the writing I've done on Economic Development Issues for freelance writing, I've done that. I can think of no other town in the state that has both functions. And those towns that do have Economic Development Directors by and large do have advisory bodies to some extent. So the model that Mary is suggesting is consistent with other communities. Is that advisory group put together by Mark or by the Board? I believe it's typically put together by the Economic Development Director. But I would presume in consultation with the town council or Select Board whatever the governing body is. And so as a advisory council, I'm not sure if I heard correctly that is it like a sort of permanent body or is it put together on an ad hoc kind of basis depending upon the kind of thing that I don't know if Mary is going on. The conversation I had with Mark would be more on an ad hoc. So for example, whoever he is having discussions with within town with different downtown events or whatnot to develop an ad hoc group to just target on that, very similar to what the Select Board has done with the ARPA council committee. You've got this set of things that you wanna get accomplished. And then after a period of time, that group dissolves and then but you keep bringing folks in. And that way I believe you keep it fresh because you will bring in different folks that have different interests versus having the same people in the room and it gets old and nothing seems to move. And I thought and correct me if I'm wrong but I thought at some point towards the middle of January because I wanted to present this in January that I think Trini had reached out to Mark to put some kind of a document together but he has not been in touch with me and I only spoken briefly with Trevor but I had been in communication with the two of them throughout the summer and as the end of November and after Thanksgiving, I have not heard from Mark or Trevor regarding this initiative. What I asked him for was to lay out how he saw it working. Okay, thank you. That was the piece that I'm missing. Like how would you do the charge for that group? How would you select? How would you that type of thing? But I haven't heard anything more from him either. Trini, what I had done is I had a very high level where I had where the economic, you know, like an organization chart where you have the economic development director as the key person. And then what I call a dotted line to the different nonprofits in town such as the RACDC and Green Mountain Economic Development Corp and so forth. And then break down bringing folks in from the community and the local businesses to participate in again, whatever initiative. We're to the question of oversight of this group and how the appointments get made. It's sort of similar in some respects from what I'm hearing Mary say to our current Arts and Culture Committee which really functions as an advisory committee. It doesn't really put forward any policies for our approval. I think so that some of what this seems to be shaping up is is when we've had different companies looking at RANDOC we've grabbed Green Mountain Economic Development. We've grabbed some banks. We've had VEDA. We've had different people that we pulled in to that conversation to try to see what makes that work. And I'm not sure that we need to have the select board approve every, it's like a working group that he would pull together to help with whatever's taking place which is part of his job duties. So I don't know that we have to go through and make a formal appointment and create a committee and do all that because I don't think it's a standing set of people. I think it's- Right, the situation is dependent on the situation. It's very much project-oriented. Yeah. And it's going to change. It's part of his job. I think the little question before us is just disbanding the formal committee. If I'm fine with it. Yeah. Seems like it's kind of complicated. Would it be appropriate to take action on this recommendation at the Reorg meeting next month? Because we- We can take action on it tonight so we don't go out and recruit people for it next month. So then they all guess what. Yeah, yeah, yeah. Don't take it personally. We're not putting anybody on this. We're getting a little bit of this. So maybe for motion. Thanks for being here. I am. I'll move that we disband the economic development council. And I will second. All those in favor? Aye. Aye. Motion carries. Thanks, Mary. Thank you, folks. I'll send what I will do is I will send an email to the council members there that are on there. And just then saying that the council has been disbanded. And I'll try to coordinate that with Mark so that it's a common message to the folks that I think their term would be up in March anyway. So it's just more as a courtesy that the council is no longer going to function the way it used to. Is that sound fair with you folks? Yeah, I think Mark should word something so they know we appreciate the effort they put in and the time some of them have been there and whatnot. Yep. All right. I will reach out to Mark on that then. That's good. Thanks. That's very interesting. I never see you on a payroll that be approved. So I know you're right there with the rest of this. Thank you. Have a wonderful evening and a good weekend. Thank you. Bye. Bye-bye. So we have two grant applications up next. The Kimball Library one, which I think was the fall broom and the recreation department applying for one through Gifford. We have no questions on those. We could move them both at the same time. I don't have any questions. I'll move the approval of both the Kimball Library and Recreation Department grants. Second. I'll move those in favor. All right. Opposed? Here's consider committee appointments. I think we had one. We have two. Oh. No. One. Third energy committee. Wendy Ross. I'm not mistaken. I'm looking at the picture on it. I don't see anything. Good night. I know I'll be on a slide. Look at this. Wendy submitted an application. It should be an application. All right. There it is. Yeah. And that's what the energy committee and the other committee appointment is for the water wastewater committee. Yeah. Which Jessam West has requested to join as a result of she's a justice of the peace and was on the board of pavement. And I see her hands with the troubles that we've had. And I'm an author to help us open the policy so we can head off some of those issues. So I suggested that she join the group. She said that she would. So without her coming forward, I would still like to have her join. Grab her while she's willing. Yes. Of course, it seems it's going to be all right. Don't let her change your mind. I think that's going to be a really important one for us to do. Yep. Especially if we go to the one flat. You're going to be here. Yeah. Send them a little complicated. Any questions on the use of those appointments? I'll move both the appointments. I'll second. Those in favor? Aye. Aye. Other business. We do not have any other business. And we do not have a manager tonight. We've got the manager report. Forge forever. Hot garbage. So we. We need to do a hurt motion on to find that we need to go into executive session. The other one to go into the session. Move that we go into executive. I will move that we go into executive session for the. Fine that we. Fine. Yes.  I'm telling you better about all this. I'll just let Tom dig in. I used to say this twice. But it's been a couple of years. So here we go. The boards. I move that we go into executive section to consider. Let's move. We consider a motion to find that executive session is necessary and prudent. And that premature general public disclosure with place of the town. And it did that. Shakespeare. And I know that I can be clear. I'll move that we actually go into executive session. Based upon. Fine. Case. I will say. All those in favor.


Hey there! Did you know that there's a difference between knock and rod knock? Try to remember some of the basics of CQC. All engines experience knock in some form, but the potential risk to your engine varies greatly depending on several factors. Rod knock is the specific result of a failed rod bearing and the sound it produces is caused by a mechanical failure that has to be physically repaired. But the other knock, sometimes called pinging, gets its name from the audible sound made by an engine that's experiencing detonation. Detonation happens as the result of improper combustion of the compressed air-filled mixture in the cylinder due to things like excessive heat, poor fuel quality or octane, valid spark plugs, overly advanced timing, and so on. So detonation and the knocking sound it produces is not rod knock, as it has nothing to do with the sound made from a failed rod bearing, but lots of detonation can lead to engine damage. Now I know that can all sound a little bit scary, but luckily most of the time it's not a big deal. Engines actually experience detonation routinely. The OEM ECU and quality tuning sources like the access port have various processes in place to respond to detonation events if they become significant enough. So if you see a little feedback knock on your favorite monitoring device, it doesn't hurt to keep an eye on it.


I love the holidays, and I really love Christmas. I love Christmas movies, Christmas music, holiday specials, and all stories associated with Santa, the reindeer, and elves will always have a special place in my heart. And if you don't know, I really love our Charlie Brown Christmas. I enjoy the festivities of decorating, and I love the warmth of giving. I love being around family and friends that I haven't seen in a while, seeing their smiles illuminated by the lights of the season. But these are my personal feelings on the holidays. Although these holidays are associated with love, happiness, and laughter, associated with family and giving, we often overlook the other side. For some, the holidays are a time where people experience higher levels of depression, anxiety, and stress over many different things over the holiday season. Those who are in school have exams that barely allow them to enjoy the holiday season for all that it is supposed to be. The cold can be unbearable, the days are ever so short, and the nights often feel endless. Finances are thin because of the season, and now you find yourself tired and stressed out. For others, it's a lonely time, where life kind of comes to a halt. Some don't have family or friends to celebrate it with. So this is a short reminder that you are not alone. There are many people out there who echo the same feelings that you do. So please seek help if you need to. Those who are strong are the ones who are able to be vulnerable. Aside from that, do the things that make you happy this holiday season. Reassure yourself of your worth. Conserve your energy for when the holidays are over. Don't be afraid to say no. Most of all, take care of yourself. For those who find themselves re-energized and warmed up by the holiday spirit. Those who find themselves surrounded by love and by care. Remember to be kind, and most of all, remember to be thankful and be grateful for what you do have. Enjoy your time. Be present in the moment. Being in spaces where you are loved and cherished are things that many people wish they had. Have fun and just relax. Happy holidays. And to those who celebrated Merry Christmas.


Towards the end of the previous lecture we started discussing about problems of finite element model updating, I provided you with the basic motivations for considering this type of problems, so we will continue with that discussion, so if you recall the question of finite element modeling arises when we deal with structures which have already come into existence, so for such type of structures one can make of course mathematical models also it becomes possible to measure the performance of the structure under operational loads or diagnostic loads, so the predictions from experimental models most often would not you know agree with predictions from the mathematical model, so the question arises how can we combine these two models, both models are prone to errors imperfections, there are various assumptions that we make in making mathematical models pertaining to constitutive laws, we may assume isotropy, homogeneity, linearity, geometric nonlinearity, hereditary nonlinearity and so on and so forth, and we postulate certain idealized boundary conditions and in structures with jointed elements we assume certain features associated with the joint behavior, so there are many and damping is another major issue where significant idealization is done, so in an experimental study none of these issues are primarily compromised, the constitutive laws joint behavior, boundary conditions, presence of nonlinearity all that are captured without any compromise, but the imperfection associated with experimental work are associated with the process of a measurement, that is calibration errors associated with sensor, the sensor structure interaction, the actuator structure interaction and problems with data acquisition that may can have problems of aliasing and so on and so forth, so both these models therefore are imperfect, so we need to somehow allow for these things when we try to reconcile predictions from these two models. Now a typical element of a finite element model updating process has as I already said an existing instrumented structure whose response has been measured under operational or diagnostic loads, by diagnostic loads we can assume that we know what the loads we are applying, and we have a finite element model for the structure on which the above measurement conditions can be simulated, then we need to have a strategy to adjust the parameters of the FE model to reconcile the prediction from FE model and the experimental observation on the system response, here we should take into account various features like mismatch of degrees of freedom between finite element model and experimental model, presence of noise in sensor, experimental work as well as imperfections in finite element model so on and so forth. There are several methods which have been developed over last few decades they can be broadly classified as direct matrix methods, inverse sensitivity methods, response function methods and certain methods which operate only in time domain and the Bayesian filtering methods. So the various scope of these methods are to examine the scope of these methods we need to also consider if the system is linear and time invariant or linear time invariant or nonlinear, nonlinear problems are obviously more difficult to handle. So what we will do is in this introductory lectures we will focus our attention on linear time invariant methods and we will not be spending time on all these methods, we will basically focus on inverse sensitivity analysis and just to give a comparison of these methods among these various methods the Bayesian filtering methods are the most powerful methods, but the background to develop these methods require you know developments in theory of probability, statistics and random processes and Bayes theorem in particular Markov process theory and so on and so forth, since this has not been addressed in this course we will not be considering questions on Bayesian filtering methods, other methods I will briefly touch upon but the main focus of our discussion will be on inverse sensitivity analysis. And we will focus on linear time invariant dynamical systems. Now when we talk about inverse sensitivity analysis it is basically inverse response sensitivity analysis, so we consider various response descriptors like natural frequencies, mode shapes, frequency response functions, impulse response functions, time histories of responses under measured or unmeasured applied loads. So the general features of a inverse sensitivity methods consist of we have a set of N generic system parameters, this could be associated with mass stiffness or damping or even forcing characteristics and all these have been parameterized and we have a set of N parameters and we have a set of NK observed dynamic properties of the system, gamma K, for example system natural frequencies, mode shapes, frequency response function, impulse response function etc. So these are obviously functions of the system parameters P1 to PN. Next we make a initial guess on the values of system parameters before measurements are made, I will call them as PU and they are again N in number. Now after the measurements are made we don't know what is the values of the system parameters from which the measurements have emanated, I call them as PDI. So if we apply a correction delta I to the initial guess that we make on system parameters we postulate that we will arrive at the unknown system parameters as indicated by the experimental results. So these delta 1, delta 2, N are the changes to be determined so that the prediction from the experiments and FE model on measured response characteristics are reconciled, so I have to clarify what this reconciliation mean, it has to be quantified, so we will see what it means. Now a simple strategy would be we consider the response descriptor which is function of the P1N system parameters and I will, for the experimental model I will expand the response descriptor around the initial guesses that we make with delta 1, delta 2, delta N being the correction. So a Taylor's expansion would lead to various terms and these gradients are to be evaluated at the, from the mathematical model with the initial guesses on the system parameters. Now this delta gamma K, suppose the first term on this side I take it to the left side, this will be gamma K PD1, PD2, PDN, minus gamma K PU1, PU2, PUN, the first term is the response descriptor that we have determined from experiments, this is the response descriptor that we have predicted from the mathematical model before we have taken the measurements. So the difference between the two is a known quantity in our work, so they can be related to these other terms in the Taylor's expansion through these terms. In a first order method we omit this quadratic and other higher order terms and I approximate delta gamma K as a linear function of delta I, so in this method we need to evaluate the gradients of response descriptors with respect to the system parameters, if there are NK number of system parameters and N number of, NK number of system response characteristics and N number of system parameters this will be a NK by N matrix. The second order method clearly also indicates the quadratic terms in the expansion, so let us consider the first order method, so we have this and this equation delta gamma K equal to summation of this gradient into the increments needs to be written for K running from 1 to NK, so I can cast this set of equations as a matrix equation with the vector delta gamma is equal to matrix S into delta where this element of SKI is this gradient, delta gamma K divided by dou P UI of gamma K, the knowns here as I already mentioned the left hand side is known here, the unknowns are the system parameters, corrections to the system parameters delta, so these are the unknowns. Now S is an NK cross N matrix to be determined from the postulated finite element model, so this would be known to us, now these constitute a set of typically over determined set of equations and this will be a rectangular matrix therefore we cannot directly invert that, so we use the pseudo inverse theory and write delta as pseudo inverse of delta gamma, so this is a matrix pseudo inverse we discussed about this when we talked about substructuring methods, so the same you know theory is applicable here also, now we can make few remarks here, if you consider this equation that a delta gamma is equal to S into delta and you are looking in this direction that means given changes that are made to system parameters I want to know what would be the change in response characteristics, this is a problem in design sensitivity analysis, for example you may like to change the stiffness of some element in a structure and you want to know what would be the change in natural frequencies and so on and so forth, so this is a forward problem and that is a design sensitivity problem. So here we determine the unknown changes in response characteristics delta gamma caused by known small changes in the system parameter, now on the problem on hand the situation is quite the opposite, here in problem of model updating the role of knowns and unknowns is reversed and we call the problem of determining delta from known values of delta gamma as a inverse sensitivity problem, the word inverse sensitivity is associated with this description, typically the number of knowns that is typically the number of knowns which are system characteristics and the unknowns which are the system parameters do not match and the matrix S is often not well conditioned, I will talk about condition number of a matrix slightly later in the lecture, it is actually ratio of the highest singular value of S to the lowest singular value of S, I need to introduce those terms we will come to that shortly. In evaluating matrix S the Taylor expansion has been carried out around the initial guess PU, now the reference state about which the Taylor expansion is done could be updated once an estimate of PD is obtained by linearization around PU, so that would mean we can set a global iteration strategy over and above this formulation where we start with the initial guess and solve this problem and get an improved estimate for delta, and that we feedback and use that as initial guess and then reiterate on this and we will get the, that is the S matrix is now linearized around an updated value of delta, and with this is solved system in a iterative way. Now to implement this method clearly we need to discuss how to determine the sensitivity matrix S, and how do you solve the resulting set of equations, there are various issues like pseudo inverse regularization, global iteration and we may like to include second order sensitivity terms in our analysis and how to proceed if we do that, so the present discussion is based on MS synthesis that is cited here by Mr. S Venkatesh, so we will first to start with we will quickly summarize the main results from linear vibration theory, suppose if you consider the undamped free vibration response of a multi degree freedom system MX double dot plus KX equal to 0, we assume all points on the structure vibrate harmonically at the same frequency, and we formulate this eigenvalue problem, this leads to set of N real valued natural frequencies and a set of N real valued eigenvectors, and they have this orthogonality properties, and we using these matrices we diagonalize this, I mean uncouple the equation of motion, and we determine various frequency response functions like receptors, mobility, accelerance, this we have considered in earlier lectures you can recall, and we can construct as well the impulse response function for the system in terms of the system modal responses, and if you want force response in time domain using modal decomposition we uncouple the equation of motion and use Duhamel integral theory and get the impulse response function, so this is we are quite familiar with what these issues are. Similarly for non-classically damped system if you recall we rewrote the equation of motion in this form Ay dot plus By equal to F of T, so that A and B were symmetric matrices, and we did free vibration analysis and obtained a set of 2N complex valued natural frequencies and 2N complex valued eigenvectors, and we showed them, we showed that they appear as conjugate pairs, the eigenvalues and eigenvectors, and the structure of the modal matrix we delineated and this was of this form, and the R matrix which is a modal matrix in this case satisfy these orthogonality relations, and using this we have derived the receptance, mobility and accelerance matrix, the impulse response functions and response to the force response and so on and so forth, so this is available we have already done, I have summarized in one place all the main results, so we'll start with study of undamped systems, that means assuming that we are going to invoke classical damping models and use these information for uncoupling the equation of motion, so the eigenvalue problem to be solved I can write it as KX equal to omega squared MX, for omega square I will write as lambda, lambda and write it as lambda MX, so I will write this as K, for ith eigenvalue I will write it as K minus lambda IM, and therefore this equation is equivalent to writing FI XI equal to 0, I can pre-multiply by XI transpose and write this equation in this form, now what is my objective I would like to derive the K and M matrices will have our system parameters P1, P2, P3, PN, and I would like to know the gradient dou lambda by dou PI for I equal to 1 to N, and similarly dou XJ that is jth eigenvector by dou PI for I running from 1 to N, so that is the objective. Now with that in mind I differentiate this equation now, so using a chain rule we get this equation, okay, so this is dou XI transpose dou PJ, FI XI remains as it is and I differentiate the second term and so on and so forth, now since FI XI equal to 0 it means the transpose of this is also 0, XI transpose FI transpose is 0, and XI transpose FI is also 0, because FI transpose is FI, it's a symmetric matrix FI, so that would mean the accepting the middle term other two term drop off and I get this equation. Now FI is basically K minus lambda MI and dou FI by dou PJ will now involve to find dou FI by dou PJ I have to differentiate this that will be dou K by dou PJ minus dou lambda I by dou PJ into M minus lambda I into dou M by dou PJ, it's equal to 0, so now I have the required quantity dou lambda I by dou PJ here and I take it on the other side I have the required expression for the sensitivity to Ith eigenvalue, this is known, this is from the mathematical model, this is known, and this you can differentiate and find out. Suppose PJ is stiffness of, suppose in a beam frame structure suppose one of the PJ is EI of one of the elements, so when you assemble the global stiffness matrix K you will be able to identify which element is associated with that parameter and you will be able to arrive at this differential, now this is sensitivity analysis, suppose how do I do the model updating with information on natural frequencies, for example I have measured natural frequencies and I have initial postulate for the natural frequencies, so from the mathematical this is the equation that we have just derived, so the lambda I I will write it as lambda I of P1 plus delta P1, P2 plus delta P2 etc, PN plus delta PN, so P1, P2, PN are the initial guesses I have dropped the subscript U and delta P are the increment that we need to find out, so this delta lambda I is given by this, so now we have derived this expression for dou lambda I by dou PJ, so this I have to use for I for the first second and N bar eigenvalues you may use first 3, first 10 or whatever it will be typically be less than the number of DOF of the system, we can assemble all these matrices in this form and we get this equation, so each term can be in this matrix need to be evaluated using this formulation, so this is the equation for the unknown increment in delta P in terms of the measured changes in natural frequencies, so this is known, this is the mathematical model, so I will use the pseudo inverse and find delta P as dou lambda by dou P lambda into delta lambda, so this is a formulation which uses only the information on natural frequencies, so you see here a term called HOT these are higher order terms, HOT stands for higher order, now how about mode shapes, suppose if I have been able to measure mode shapes and I am able to predict mode shape from my postulated FE model I would then know the difference between the measured mode shape and the predicted mode shapes, so how do I get sensitivity with respect to mode shapes, so to be able to do that we will assume that mode shapes are mass normalized, so I will consider a pair of eigen solutions with indices I and S, so I have XI transpose M XS is delta IS, delta X is the chronicle delta function, now I will differentiate this with respect to PJ I get this equation, now by noting that these quantities XI transpose M and dou XS by dou PJ and these are scalars, so I can reverse the way in which they are written and I can rewrite this equation in this form, the quantity that I am looking for are this dou XI by dou PJ and dou XS by dou PJ that we have to segregate and find out, now similarly we have other orthogonality relation XI transpose K XS is lambda I delta IS, so again I will differentiate this and we get one more equation, now we also have the basic equation FI XI equal to 0, this is an eigenvalue problem, from this I get the, if I differentiate with respect to PJ I get XI and XS I get two equations as shown here, so now I have four equations I can club all of them, so what I have used is the basic statement of eigenvalue problem and the two orthogonality relations, so I have Ith eigenvalue and Sth eigenvalue, therefore I have two equilibrium equation, one for I and one for S, from that I have got these two equations, but I also know XI and XS have certain orthogonality property, namely it is XIs are orthogonal to mass matrix and XIs are orthogonal to stiffness matrix, right, so I have used these four equations basically to derive the gradient, so this I cast it in this form, so that is this equation has been derived based on considering two eigen pairs here, and this is dou XI by dou delta J dou XS by dou delta J is equal to this. Now why to consider only two eigen solutions, we can consider three eigen solutions, so we can repeat the whole story, I can get, while considering three eigen solutions I will write the equilibrium equation for the three pertaining to three eigen solutions, and there will be orthogonality relations with respect to mass and stiffness between three is three eigen solutions, so if I combine all that I get a larger set of equations, so here I am considering Ith, Rth and Sth eigen pairs, so this can continue, so I can consider IRSK and get a much larger set of equation, so somewhere you have to stop with this formulation and that becomes one of the algorithmic parameters in implementing the method, so now therefore based on this I have now equations for using this of course I can again consider the changes in mode shapes at various coordinates and for various modes, and again get an equation of the form this, where now this delta gamma will consist of changes in the values of mode shapes between the what is measured and what is predicted, and yes is the, these matrix consisting of these gradients which have to be determined using one of these formulations, and delta P is a change in parameter that we wish to do, so we have now changes in natural frequencies and changes in mode shapes, so all of them can be clubbed, and we can write now if we consider N bar number of modes I have results on N bar natural frequencies and N bar mode shapes, and those mode shapes themselves will be measured at several points, so that also has to be understood, so there will be issues about sizes of this equation, suppose if I have R bar number of mode shapes included, and S bar number of special points where the mode shapes are evaluated, this delta gamma will be of this size, S will be of this size, and delta P will be of this size, so the final equation to be solved is this, and again I get delta P as S pseudo inverse delta gamma, so in this approach what we have done is therefore we are considering only natural frequencies, then eigenvectors also, and then while formulating sensitivity we may consider 2 eigenpairs at a time, 3 eigenpairs at a time, 4 eigenpairs at a time, so on and so forth. Now this analysis was for undamped natural frequencies, but in an experimental work what you measure will be often invariably the damped natural frequencies and damped normal modes, there is no way you can eliminate damping in an experimental work, so what we measure in a laboratory is always, even for free vibration characteristics it is always damped natural frequencies and damped normal modes, if you recall we have discussed the nature of these solutions and we have shown that the natural frequencies and mode shapes will be complex valued and they appear as complex conjugates, so if we consider equation of motion in this form we write this equation AY dot plus BY equal to F of T, now A and B are the new structural matrices, they no longer have the direct interpretation of being either mass or damping matrices, A has in fact M and C, B has M and K and so on and so forth. Now the free vibration solution if you want to consider it will be AY dot plus BY equal to 0, that is again a set of constant coefficient linear homogeneous differential equations, therefore an exponential solution would be acceptable and that leads to this eigenvalue problem, so it will lead to a set of 2N eigenvalues where the eigenvalues appear in complex pair and eigen solutions also appear in complex pairs and we have this relation and this R modal matrix R has these two orthogonality relations. Now again we can consider sensitivity of only the eigenvalue, suppose you consider the Rth eigenvalue with the governing equation will be A omega R, RR is minus BR, so I will write FR as A omega R plus B that leads to FR into RR equal to 0, so I will pre-multiply by RR transpose and write this, I differentiate this with respect to PJ and use the fact that this is 0 therefore these are also 0 and I get this equation, so the formulation proceeds on exactly the same lines as we did for undamped system but of course these quantities are complex value, now since FR is A omega R plus B doh FR by doh PJ can be evaluated from this I get this, now again using the fact that some of these are scalars etc. I will be able to write this, now since RR transpose ARR is 1 I get doh omega R by doh delta J as this, this is the gradient that we are looking for, so if you are going to use only changes in natural frequencies as for updating you get this updating equation, these are changes in delta P, these are changes in the natural frequencies complex valued and this is a gradient matrix that you have to evaluate from your FE model, more compactly I can write it as delta gamma as this, now this is a complex valued quantity therefore I can separate the real and imaginary part I will write U plus IV and delta omega itself are W plus IZ, so substitute here and separate real and imaginary parts I put it here and separate I get these equations from this I can get delta P as this equation, okay, so this is the updating equation that we have to use, now if you want to include more shapes the story is same again you can consider 2 eigen pairs, 3 eigen pairs, 4 eigen pairs so on and so forth, so again just for illustration we will consider one instance, RI transpose ARS is delta IS from this I get this equation by differentiating and noting that these are true we simplify this and I get this equation, then similar equation I get with respect to the other orthogonality relations this is that equation, now I have statements of eigenvalue problem for Ith mode and Sth mode, starting from that I get other 2 equations, so the 2 equations for Ith and Sth mode emanate from the statement of the eigenvalue problem and other 2 from the 2 orthogonality relation, so these 4 I will club and obtain a over-determined set of equations for gradients of eigenvectors with respect to delta J, so I can again do a pseudo inverse and find this quantity, so if you consider 3 eigen pairs this is it, so you will write 3 equations for eigenvalue problem and 3 eigen, the orthogonality relations among the 3 eigen pairs with respect to A and B matrices, so all those equations if you club you get this equation, 4 eigen pairs the equation becomes more complicated, but all these terms will be available to you you can do this, again if you combine all the equations for updating obtaining the final updating equations if you have data on N bar natural frequencies and N bar mode shapes evaluated at some S bar number of points and so on and so forth, if the final equations can be written as delta gamma is S delta P and various sizes of these quantities as before are delineated here and you should notice that these are state space form therefore dimensions will be 2 into N bar then what about it was earlier, so we get the final equation in this form, again delta P is S pseudo inverse of delta gamma, so since they are complex valued to facilitate computation I can separate real and imaginary parts rewrite the equations and I get the final updating equation to be this, this is with respect to natural frequencies and mode shapes which could be real valued or complex valued and you can include as many number of modes as many number of orthogonality relations as you wish and develop these methods. Now if you look into the experimental modal analysis literature the primary quantity that we measure in an experimental work often happens to be either the impulse response function or the frequency response function, from the given frequency response function a matrix of frequency response function we extract the natural frequencies and mode shapes, so if you don't want to do that extraction you want to deal directly with what has been measured, I mean even that FRF is processed from what we measure but that is relatively a more primary quantity than the secondary quantity like natural frequencies and mode shapes and damping and so on and so forth, so if you want to now perform a sensitivity analysis on frequency response functions itself, so you can predict the frequency response function from your postulated mathematical model and compare directly with the measured frequency response function, so you will get certain differences and that you can now study to determine piece, so how do we do that, so we have now several descriptors of frequency response function, this we have again seen in earlier lectures we have receptance, mobility, accelerance, we have described all this earlier these are all complex valued quantities. Now let us consider for purpose of illustration receptance matrix, so let's assume to start with that it is a square matrix, now I can write alpha of omega into D omega is equal to I where D is the inverse of the receptance matrix, now I want to differentiate what I want to find is dou alpha IJ with respect to dou PK, some IJth element evaluated at some frequency omega with respect to some PKth parameter, so I basically I am interested in this gradient, so differentiate with respect to PJ I get dou alpha by dou PJ into D plus alpha into dou D by dou PJ equal to 0, so if you solve for dou alpha by dou PJ I get this equation and this is minus alpha dou D by dou PJ into D inverse, D inverse is nothing but alpha, D is we can now differentiate D and find out this quantity, now therefore I have got dou alpha by dou PJ as given by this, now suppose if we have measured alpha RS of omega for say R equal to 1 to NR and S equal to 1 to NS we will get a NR by NS matrix and this omega could again vary from 1 to some N omega number of frequencies. So if I assemble all the observed changes there will be related to the unknown changes in system parameter through this matrix, so the sizes are spelt out here and you can easily imagine these matrices will be now very large sized because for every frequency omega and for every alpha RS of omega you are writing one equation, so then the number of equations to be solved can be excessively large in relation to number of system parameters to be determined and this can pose considerable computational difficulties. Again we can separate the real and imaginary parts because we are dealing with frequency response function therefore they will have real and imaginary parts, so you have to separate and you can get a similar equation as we got earlier. Now let's quickly foray into what happens if we consider second order terms in our Taylor's expansion, so it is easy to explain that with respect to FRF sensitivity, so we got this first order sensitivity this is exact, and if I now differentiate with respect to PK, I have done it for PJ now if I differentiate with respect to PK I need to simply differentiate these terms, so I get this, so this is straight forward to be evaluated, so evaluation of this first order and second order sensitivity for FRF's presents no difficulties, it can be done in a straight forward manner. So let's now consider change in observed parameter using first order terms and second order terms, these are a set of over determined non-linear in this case it is quadratic algebraic equation, we can use an iterative strategy to solve this equation, so what we will do is we will start an iteration count Q and at the Qth iteration step for the second order terms I will use the, here I will use Q plus 1, here I will use Q, so we can iterate this and find out the solutions. So we can write this as this equation matrix form S delta Q plus 1 is delta gamma minus S2Q, so where S2Q is the second order gradients evaluated the previous step of iteration, so to start the iteration we can use a first order analysis, so you can do this and then start the second order iteration, so this approach is likely to lead to large number of equations a few unknowns and this may pose numerical difficulty as before, but the advantage of second order sensitivity is if your initial guess is far away from what is the true value it gives you a greater margin of error between your prediction and the true value because you are including quadratic terms in your expansion, so this is an advantage of this method, later on with through some numerical examples we will be able to see this. Now we have looked at now natural frequencies, mode shapes, FRFs, are there any other descriptors that we can look at? Now FRF matrices in experimental work are often rectangular, what happens if we perform a singular decomposition of those FRF matrices, what are the quantities that we get? We have seen that the equation for FRFs if you use there will be very large number of equations to be solved for the updating system parameters, so can we you know effectively do some kind of data reduction by instead of considering all the FRFs can we simply condense the data by considering singular values and singular vectors and so on and so on, there is a type of question. We need to start with some preliminary, we discuss what is known as complex mode indicator function, so before that as a precursor to that we need to discuss what is singular value decomposition, so I will quickly run through this, so let A B A N cross N non-singular matrix and consider the eigenvalue problem A X equal to lambda X, so we will consider the situations where I get N eigenvalues and N by N eigenvector matrix and I will normalize the modal matrix so that phi transpose phi is I and phi transpose A phi is this diagonal matrix of eigenvalues, so the eigenvalues and eigenvector therefore satisfy the relation A phi is phi lambda, and now pre-multiplying, post-multiplying by phi transpose I get A to be phi lambda phi transpose, so what I am doing is I am decomposing A in terms of an orthogonal matrix phi and a diagonal matrix lambda, so this representation is very useful in evaluating for example functions of A and so on and so forth. Now the question we ask is what happens if A is a rectangular matrix, can we get a similar type of decomposition for a rectangular matrix? Obviously we cannot talk about eigenvalues and eigenvectors of A directly, so what we do is we define two matrices, B is A into A transpose, suppose A is M cross N, B would be A, A transpose will be M cross M, and B transpose will be N cross N, so we can do eigenvalue analysis on B and B transpose, okay, and we can find the M by M modal matrix for matrix B and that will have this orthogonality relation. Similarly Q2 be the N by N eigenvector matrix of B transpose so that Q2 transpose Q2 is high, we can show that the nonzero eigenvalues of B and B transpose will be identical, see B will have M eigenvalues, B transpose will have N eigenvalues, but there will be certain rank deficiencies associated with these matrices, the nonzero eigenvalues of B and B transpose can be shown to coincide, in fact we will be able to write that A is Q1 sum sigma Q2 transpose, where sigma is a N cross M diagonal matrix of square root of the nonzero eigenvalues of B and B transpose, we can verify that for example A is Q1 sigma Q2 transpose, suppose I am a post-multiply by A transpose and use these definitions I will be able to show that A A transpose will be Q1 sigma sigma transpose Q1, so this is the kind of decomposition for B that we have just now discussed for a square matrix, similarly A transpose A which is N cross N I will get a similar decomposition, now what is of interest is A itself can be decomposed like this, this is known as singular value decomposition of matrix A, and this Q1 and Q2 are known as singular vectors, Q1 is a left singular vector and Q2 is a right singular vector, and sigma is the singular values of A, we can see a quickly an example suppose A is a 4 cross 2 matrix, we can find out the sigma matrix that will be this, and Q1 and Q2 will be this, and you can verify that Q1 transpose Q1 is I, and Q2 transpose Q2 is I, and if you multiply now Q1, Q2 transpose we will get this matrix which is nothing but A, you can verify that, this is just an illustration of what I am telling, we will now consider the question what will happen if I now perform singular value decomposition of the FRF matrix itself, so then that leads to what is known as complex mode indicator function or CMIF, so let us consider NR cross NS FRF matrix alpha, it's rectangular, we will define B as alpha into alpha H, and Q as alpha H into alpha, where the H is the conjugate transpose, now B will be NR cross NR, Q is NS cross NS, so B and Q are real symmetric with real eigenvalues, so the spectrum of these eigenvalues are called complex mode indicator functions, what they are, to understand that we will consider a simple example, I will consider a 7 degree of freedom system with mass matrix as this and stiffness matrix as this, now you can, if you compute the natural frequencies you will be able to get these natural frequencies, you will see that all these 7 natural frequencies are distinct, now if I plot the CMIF, this is FRFs, suppose for one row of acceptance functions I show these have 7 peaks corresponding to 7 natural frequencies, now if you plot the spectrum of singular values for the system, if you see the blue line, the blue line is a spectrum for the first singular value, and you clearly see 7 peaks which correspond to the 7 natural frequencies of the system, so no problem here, now we will change the system slightly, we will alter the mass and stiffness matrix, now I have a very peculiar system in which there are again 7 natural frequencies, the first natural frequency is 10, but all remaining 6 natural frequencies are 28.284th, that means the remaining natural frequencies repeat 6 times, so now if you compute the frequency response function you will see only 2 peaks, it appears as though you are dealing with a 2 degree freedom system, so this FRF matrix will not show, FRF plot will not show that some eigenvalues are repeated, but on the other hand if you plot the spectrum of singular values you will see that there will be, if you plot the singular values for the plots of 7 singular values you will see that there will be 7 places where they speak and I obtain peaks for the 2nd, 3rd, 4th, 5th and 6th at the frequency 28 thereby indicating that the frequency 28.2843 etcetera is repeating 6 times, so this is used in industrial experimental works to characterize repeated natural frequencies or closely spaced natural frequencies, so this is a very useful tool. Now motivated by this we can consider problems of inverse sensitivity of a singular values of a FRF matrix, we can do a single inverse sensitivity analysis with respect to CMIF itself for example, so we will again consider this we have introduced these notations, now B and Q are these matrices, I will consider the eigenvalues BX problem with respect to B and I have these relations and you must notice that when I talk about FRF it is at its value, there is a frequency, driving frequency parameter implied in this, so all these analysis has to be done for every frequency, so the driving frequency is now fixed and there could be N omega number of driving frequencies that has to be borne in mind. Now I can do the, now for B matrix and this Q matrix I can do the eigen sensitivity analysis whatever I did for natural frequency in mode shapes etcetera, so I get, I will not run into these steps we get by analyzing B matrix I get certain equations with eigenvalues alone, eigenvectors also, this is eigenvalue equation with only eigenvalues we can focus only on singular values, we will not include singular vectors in our discussion, so this is the equation. Now if you write these equation for N omega number of driving frequencies I get a set of large set of equations as shown here and these are the updating equations that can be used, okay. So we will see that this helps us to deal with repeated natural frequencies, when I considered the derivation of eigen sensitivities the question of possibility of eigenvalues repeating was not addressed, so if you have a system with certain symmetries so eigenvalues could repeat, so in that case how do you do updating, because gradients of natural frequencies for frequencies which repeat involve certain additional considerations. Now what we have done is the generic form of no matter which respond descriptor you use, the generic form of the equation has been delta P is equal to some S plus delta gamma, now is this solution strategy always workable, is the next question we have to consider. So actually it turns out that it is advantageous to refine this solution strategy by using what is known as regularization that is Ticano regularization, what it means is what I am going to explain now. Consider this set of equation AX equal to B, A is a square matrix M cross N, B is M cross 1, X is N cross 1. Now we define condition number of A as the ratio of the largest singular value of A and the smallest singular value of A, now before I proceed I can take a simple example, suppose if I take A to be a matrix 4 cross 6 matrix of 1's, the condition number A is not defined, because the lowest condition number is 0, it's rank deficient so there will be a problem. Now what I do is I add small perturbations to this, so this is the A matrix now, okay, and condition number becomes 1629.4, so if in your analysis if this is a matrix that you have to deal with but because of perturbations errors and so on and so forth you observe this then if you attempt to invert or solve these problems, fine pseudo inverse etc you are dealing with a highly ill conditioned matrix, so I will show some more issues related to this as we go along, clearly condition number of identity matrix is 1, this you have to, now what we do is instead of considering AX equal to B we consider a modified version of this, for example how we proceed to find pseudo inverse I pre-multiply by A transpose, and A transpose A is a square matrix that I will invert, that is what we have been doing, that is I have AX equal to B, I can pre-multiply by this, so this is N cross M, this is M cross N, this is N cross 1, and this is M cross 1, and A transpose is N cross M, so I get N cross 1 equation and I can invert this matrix and find X, that is what is our definition of pseudo inverse is. Now I don't want to do that, what I will do is I will introduce a additional term XI into I, this XI is a scalar parameter, now instead of inverting A transpose A I will invert this matrix, okay, this is one as regularization parameter to be selected such that we improve upon the condition number of this A transpose A matrix, okay, now what that means? Suppose from this I get X as A transpose A plus XI I inverse A transpose B, we can show that this solution is equivalent to minimizing the quantity AX minus B modulus plus norm plus XI into norm of X, now what is A norm of AX minus B? It is a error norm, okay, if for a given value of AX AX minus B must be equal to 0 but you are not getting that, so this norm is actually a error norm, on the other hand this norm of X is a measure of smoothness of the solution, if there are two alternate solutions one which is smooth is what I prefer, that means if elements of AX oscillate too much that is highly non-smooth type of solution, whereas all elements are close to each other then the norm of that matrix will be less. Now if XI becomes arbitrarily large, how to select XI is still the question that we had to answer, see we cannot go on increasing XI indefinitely, then you will be fiddling with the physics of the problem, you will be altering that, that is not acceptable, on the other hand if you put XI equal to 0 then you are back to the problem of inverting a ill-conditioned matrix, so there is obviously a trade-off in selecting XI between the values of this norm and this norm, so what is done is we consider what is known as a L curve, and there is a useful reference I have given here you can see that this is available on the web, let us consider a simple example A is this matrix, now I will do a singular value decomposition of this and I get these three, this is a left eigen, singular vector this is the singular value and this is a right matrix of right singular vectors, we can show that condition number of A is about 1097.5, now you consider AX equal to B where B is given by this, now if you find A pseudo inverse B I get answer as 11, which is a nice solution, now let's consider now we will add a slight noise to B, B is 0.26283.3, I will make it 0.27, 0.25 and 3.33, this is quite conceivable, in experimental work such type of noise is quite possible, now if you now find using the same formulation X becomes true answer is 11, it becomes 7 and minus 8, that means in this type of calculations are unforgiving as far as noise is concerned, a slight noise can distort the answer, this is a very eloquent illustration of that, this happens because of, this happens because the condition A matrix which we are trying to invert A transpose A matrix has a very large condition number, right, so what we do is we now instead of solving that problem I consider A transpose A plus Xi of I X equal to 0, now how this Xi is selected is we plot the two, you know scalars A X minus B norm and norm of X for different values of Xi, and the point which is closest to the origin which happens to this typically turns out to be L shaped curve and at the bend the point is taken as a optimal value, so this can be programmed and we get the Tikhonov regularization parameter 0.03, if I use that now, I get now the D matrix, A transpose A plus Xi is this and if I do a singular value decomposition I see that condition number is now 190, it has dropped from nearly 1097 to 190, the solution I get is 1.19 and 0.70, so this is lot more acceptable than 7 and minus 8, so what we have to do is every step where we are solving word determined set of equations we have to do a regularization, that is always helpful, so in summary now what we have done is where the updating equations have this form, delta gamma is S into delta P and we use regularization and find delta P, and we select Xi by using the L curve approach and delta P leads to this, on this we will impose a global iteration, that means I'll start with the initial guess on P and I will evaluate this S matrix at that value of P, I'll solve this and find the increment to P, and I will now revise my S matrix instead of evaluating at the original value I will evaluate at the upgraded value, so this iteration I will continue till some norm on delta P converges. Now a refinement on this would be to introduce a second order terms in the Taylor's expansion, so again we can retain all these ingredients, regularization, global iteration, all these steps can be introduced, this delta gamma as we have seen we have used undamped natural frequencies and mode shapes, damped natural frequencies and mode shapes, FRFs, singular values of FRF matrix, you can of course include singular values and singular vectors of the FRF matrix, and the question on at what frequency you would like to include these rises that can be handled, there are few issues associated with that, so we can close this discussion by making few observations, what happened to measurement noise in this? There is something interesting here in the sense when we use FRFs, FRFs are typically obtained by averaging across several measurements, so to some extent the measurement noises, effect of measurement noise is mitigated when you use averaged FRFs, okay, so noise is eliminated by averaging, so that is one place where we explicitly handle presence of noise, but as far as imperfections in the mathematical model itself is concerned, this there is no explicit model for the imperfections, so the answers we get on delta PR deterministic in this approach, this delta gamma if you use undamped natural frequencies and mode shapes large data gets compressed, large data set comprising of FRFs gets compressed to few scalar numbers and few functions, that is the mode shapes and few natural frequencies, similarly this is also true if you are dealing with damped natural frequencies and mode shapes, again there is a data compression, but on the other hand if you are using FRFs you have to deal with very large amount of data, similarly singular values of FRF matrix there is a data compression, again singular values and singular vectors of FRF matrix if you use again there is a compression of large data, now actually if you use singular values and singular vectors of FRF matrix you can show that the inverse eigen solution method will be a special case of this approach, I am not sure if you will be able to get into all the details, but I am just pointing out you can explore that fact if that is true by your own methods, now in the next class what we will do is we will consider few examples and illustrate the updating method that we have discussed in this lecture, so at this point we will close this lecture.