CHAPTER 01.01: INTRODUCTION TO NUMERICAL METHODS On Solving an Engineering Problem - Part 1 of 2   In this segment we are going to talk about what are the steps in solving engineering problems.   This is part of introduction to Numerical Methods. Numerical Methods are going to be used by you as an engineer or scientist to be able to solve a problem. So let's look at an example here. So first let's go and look at what the steps of solving an engineering problem are. First thing which you have to do is that you have describe the problem itself. That is the first thing you have to do because if you don’t know how to describe the problem you won't be able to solve it. So it is very important that somebody is giving you a problem that you need to first write a description of it, what it is and what we are looking for. Then, what you may do is you might develop a mathematical model for it and some people might argue that hey, you have to develop an experimental model for it. There is nothing wrong with that. It all depends on what kind of approach you are taking to solve the problem so you may either have to develop a mathematical model or an experimental model to be able to do that. Eventually, even an experimental model has to be, you have to use some mathematical model to be able to solve the problem or to be able to present the problem. Once you have the mathematical model, so we are restricting the application to our numerical methods, what we are going to use in numerical methods. Then you want to solve the problem, so you have to solve the mathematical model. Now, the solution might involve analytical means, numerical means, or maybe using some kind of a package program and once you have solved the mathematical model, what you have to do is you have to use the solution. Many times people, students think that once they have solved the mathematical model they are done but that is not what your employer or anyone else is looking for. They are not looking for just the solution of the mathematical model, but also that you implement that solution so that the problem is solved.   So what I am going to do is, this might seem to be very descriptive of what I am trying to tell you, what I am going to do it I am going to take a real life example and show you these four steps of solving an engineering problem. So here what you are seeing is a picture of a bascule bridge up in St. Augustine and as you can see here that the bascule bridge is simply like a first class lever. So you have a fulcrum right here and what you have is like a seesaw. In fact, the word bascule comes from a French word called bascule and it means a seesaw, so basically what is happening is that it is opening the bridge. So when you have a counterweight here, so you have a counterweight here and it just opens the bridge. When you let the counterweight go it basically closes the bridge. So what does that have to do with an engineering problem is as follows. This is when a bascule bridge is being assembled together. So this is a bascule bridge that is over our own Hillsborough County River bridge in Tampa, Florida. And when they were building it we took a picture of it and what I want you to concentrate on is on this part right here which is the fulcrum itself, so that's our fulcrum. When we say bascule bridge THG, what we mean by T, T is the Trunnion, H stands for Hub and G stands for the Girder. So you have a Trunnion-Hub- Girder assembly that makes the fulcrum of the bridge. And once we get a closer picture of this it will be more evident what we mean by that. It is as follows: If we take a closer picture of here, this is what you are seeing here. That is the Trunnion, this part. It is a hollow steel shaft which can be, in this case it is about a foot in diameter. And this is the Hub. The Hub is attached to the girder of the bridge on which you want to put the span of the bridge. So, that is why it is called the Trunnion-Hub-Girder assembly because you've got a Trunnion, you've got a Hub and you've got a Girder in which it is being assembled. Now, the way to assemble it is that, what you do is you take this trunnion for example and you cool it down so it has to shrink fit into the hub here. So that is the first part of the assembly that you do. You take the trunnion, it's a separate part, a hollow steel shaft. You will put it into a liquid medium such as dry ice and alcohol or in liquid nitrogen let's suppose and then what you are going to do is, you are going to since it shrinks, you can shrink fit it into the hub itself which is this part here. Let's go through that and see how that works. So, these are the steps which are used to be able to make the Trunnion-Hub-Girder assembly. So you have, this is the Girder, this is the Hub and this is the Trunnion. So what you are going to do is you are going to take this Trunnion here and you are going to immerse it in some kind of liquid such as dry ice, a cooling medium such as dry ice and alcohol or liquid nitrogen. And you are going to cool it down. Once it cools down then the diameter contracts and you are going to put it into the Hub and that's what results in the assembly here. Now what happens then, in order to be able to put it in the Girder, which is this part right here. In order to be able to put it in that Girder what you have to do is take this Trunnion-Hub assembly, the Trunnion-Hub which are together. You're going to take that, put it into liquid nitrogen or dry ice and alcohol mixture and cool it down so that the Hub now goes into the Girder and results in the assembly called the Trunnion-Hub-Girder assembly as you are seeing in the bridge right there. So, what was the problem? That's what we want to be able to see. The problem is right here, that on one of the bridges what happened was that as you were trying to put the, as you cooled down the Trunnion and you tried to put it in the Hub it got stuck. Luckily, in this case they were able to take it out before it got stuck for good. Now, what can happen is that, if it would have gotten stuck and the trunnion could not have been taken out it would have cost about fifty-thousand dollars to get a new one. Plus maybe a month of delay in the construction because one has to make a new one and be able to re-assembly it.   So, why did it get stuck? Let's go and see why did it get stuck. Now the magnitude, according to the specifications there are some specifications based on something called FN2 and FN3 fit. We needed a contraction of 0.015 inches. So, that's the amount of contraction that was required to be able to have in Trunnion before it was put into the Hub there. So the question arose is that, hey, let’s go and see that whether we did this contraction of 0.015 inches before it was put in the hub. Maybe that's why it got stuck because we didn't have enough contraction. So, some calculations were done. Some calculations were done right here is that, let's go ahead and use a physics formula here to calculate the amount of contraction which will take place. So the contraction diameter is simply given by the diameter of the Trunnion which is this part here, the diameter of the Trunnion which is here. And then multiply by the thermal expansion coefficient if steel, multiplied by the temperature change, delta T. So, what they were able to do it that the diameter was about a foot as I was telling you, it is about a foot in some medium size bridges. And then alpha we took from a handbook for steel at room temperature and then we took delta T. The delta T is the difference between the temperature which you have at room temperature which is 80 degrees Fahrenheit, let's suppose and minus 108 is the temperature of the dry ice and alcohol mixture. So, the delta T difference you are getting is minus 188 because that is the change in temperature which is taking place. So, just simply plugging into the simple equation, it's pretty harmless. You've got diameter times alpha times delta T and that's what you get as the contraction. Now, as you are seeing here that the contraction is more than 0.015 inches as required by the specifications. So people thought, the consultants thought that hey, there is nothing wrong with the amount of contraction which we got because these calculations are telling us that we should have gotten a contraction of 0.015 inches or more, as required by the specifications. And some people thought then maybe it was not allowed to be cooled enough when you immersed it in the dry ice and alcohol mixture, that it was not allowed to be there, to reach steady state and that's why it got stuck. Let's go and see if that is really true.