Modeling Engineered Systems - Demo - Week 3

[MUSIC]. Hi, my name is Andy Wilson and I'm a graduate student working in Todd Murphey's lab. And I'm Henry Hung, I'm an undergraduate working in Todd Murphey's lab. >> And I took the class that this one is based off of last year. >> So we're going to be going over constitutive laws in this demo. And explaining how you at home can work along with us and, using some basic household items, can test spring constants, find damping coefficients, and one other constitutive law that we're going to explain in the end. >> So to do this, you're going to need a few basic items and, Henry's going to talk about how we set up our system here. >> So for our system, we will have spring mass system, which, modeled by this diagram, spring and a mass. Now I will be using a Slinky to both simulate The spring and the mass. The spring will be here and the mass will be the bundled, taped part of the bottom part of the slinky here. >> So in case you don't have a slinky at home, you can use a variety of other objects, for example a rubber band if you have anything from the grocery store. or even a hair band you have laying around the house. Just hang the rubber band off of something like a table and hang something of some weight at the end of it. It could be an apple for example. and Henry will also tell you how an apple may be useful later on in this lecture. So, you can also find a variety of different types of slinkies at the store. There's a little slinky you can also use. >> or any sort of spring that you have lying around the house. Anything that has some elasticity we'll we'll try to find the spring constant for. And to do that, Henry's going to explain how you go about finding the spring constant. >> Okay, as Andy said will be finding the spring constant. As you've learned in constant, the spring constant is modeled by this equation F=kx. When F is the force, K is spring constant, which we're trying to find. And X is the stretched length of the spring. Now, to find the spring constant, we need to find what F and X are. And because you'll be hanging the spring off of like a table or something, F would be modeled by newton's equation. Goes to Ma, as I said before you're hanging it off. So A, in this case, will be just be gravity, goes to Mg. Now gravity is a universal constant. Which will be 9.8 meters per second squared. And, and now in this lab, it doesn't have to be that accurate. So we can estimate g to make all math and life easier. To be about 10 meters per second squared. Now we know what G is. And we need to know what M is. Now, as i said before, M is modeled by this bundled, taped part of the slinky. And you can be using, like, as Andy said, an apple to be your mass. So, first you have to figure out what the mass actually is to find, plug it in. So of course the most accurate way to find a mass is like, get a scale, mass it, and just plug in the number. Of course in this demo, it doesn't have to be that accurate and you might not have a scale on hand. So there are different ways to estimate the mass of some, of an object. For example, one is like, to cold mass it. Just hold it in your hand like, say like, And this feels about, I don't know, 100 grams, 1 Newton, 1 pound, something like that. Of course that's not, you know, too accurate. Especially if you're not good at it. So another way, and slightly more accurate, is to compare it to something. For example, take this apple. Now, this apple weight about 100 grams or, at least according to Internet. And it can just compare it to the Slinky. Say like, oh, this Slinky is about twice as much as the apple. So the Slinky'll be about 200 grams. Now after you mass it, you have to mass your mass. Whatever is, using for your force. You have to write it down. In my case. I previously massed the slinky to be about 220 grams. Now, because only half of it is being used as the, as the mass in the system, that means the mass will be about 110 grams, or, in kilograms, it will be 0.11 kilograms. Now, as a side note, it's always good to have your units constant or consistent. I'll be using SI units, which will be meters, seconds, and kilograms. See, now we know what f is. And we need to know, now we need to know what x is. X. So now what x is, you first have to hang, your, spring, or whatever you're using, off like, a table. Now, in this case, x is the stretch length. So you have to figure out the difference between the unstretched and stretched length. Now for this Slinky, the unstretched length will be like this. When it's, you know, not being - when a force is not being applied to it. Now the stretch length will be the difference between this and when a force is applied with it, so put your force and just let it hang. It may take a while to make it stop, you know, oscillating like this, so wait a couple moments or try to stop it. So now the stretch length will be the difference between this and this, which you should, take a ruler, and just measure it, to be a, this is approximately 26 centimeters. Or, to keep our units consistent, convert to meters to be about. 26 meters. Now, now we know what F and x are, so we can figure out what k is. Now, convert, isolate k on one side, to have k equals to F. If I write X or, as I said before f equals to mg over x. Now we know where m is, in my case, it's. 11 kilograms, g, ten meters per second squared, x,. 26 meters. Now your, of course your values are going to be different, except for G, because using different systems, or different objects to model the systems. Now, I just plug in the numbers and I get k (or k1) to be about 4.223 Newtons per meter. Now, of course in a science in engineering, it's probably better to do the experiments more than once, just to make sure you're doing it correctly. As, I may have made a mistake in that note. So, to do this experiment again, first we have to change the force, or in this case, change the mass. Which will then change the x value, and hopefully because we want. Our system's to be linear, that k will be approximately the same. Now I did this again. I instead, but instead of bundling half of the Slinky, I bundled a third. Now, I'm not going to do this now but I want to show you the value I got. Which I got k2 to be about 4.4 [UNKNOWN]. So try, so just try to do the experiment again, but of course change the force. Now you may say like your value and my values are going to be different, you say oh but this is not, that means it's not linear, right? Well, no because ideally engineered systems are linear, but of course, in real life. Things aren't ideal, like manufacturing errors. Of course, the big one in my case, measuring errors because I estimate a lot. So that accounts for. 17 Newton Meters. As long as your values are close that pretty much is, that pretty much says that you did it correctly. >> And now, spring constant is only one of many constituitive elements. And now Andy will show you more about constituitive laws. >> So as we saw in Henry's, demo. We already have the spring and, mass parameters for our spring mass system. Now the next constituitive law that we're going to take a look at is damping. Now in our system, or whatever object you may have, there's probably some sort of damping present in the system. And to show you that, I'm just going to draw a line right down here and we're going to take a look at what the spring does if I stretch it out at this point and just let it go. Now as you can see, the spring is pretty quickly losing. A lot of its energy and not returning back down to the point that we had drawn previously. So, that should indicate that there's some loss of energy and that is due to damping in the system. Now as we saw in class, the damping linear damping is also called viscose damping. And we're going to be using that constitutive law to try to determine what the damping coefficient is for our spring mass system. So we're going to do a slight alteration to our setup here. We're just going to add in a damper. Right in parallel with the spring. Now the f-, damping force is going to be opposite our x direction, so it's going to be pointed up, and I'm going to call that Fd. So now that we've set up our diagram, we can try to derive the equation that we'll need. to create a simulation of the spring-mass-damper system as the next step. So, as you've seen in class, Fd equals b times x dot. So we have our damping coefficient that's directly related to the velocity of the end point of our spring. Now using this constitutive relationship we can add to the equation that you had in Henry's part of the demo. And we're going to be moving the slinky. So, we have to take acceleration of the mass into account. So, the overall equation for this is going to be mg minus kx minus bx dot equals mx double dot. Now, make sure that you take a look at this equation, and try to drive it for yourself uing the class notes and lectures from class. And it's pretty simple. All that we're really adding is this damping term, and our accleration. So now that we have the equation it may be clear that the damping coefficient is a little bit of a different parameter than either the mass or spring constant in terms of measuring and figuring out what it is. And that's due to the fact that we have to move the spring in order to figure out what damping is going to be. If we just let this spring sit here, we can take a look at it, we can measure the length and everything you did in the first part of the demo to figure out the mass and spin constant. However, dampening we can't figure out just by looking at a static system. We have to add some dynamics to it. So to do that, we can either stretch out the slinky, or we can do some other sophisticated techniques, and we actually do some of this in our lab. It's called system identification. And, we're designing different ways that you can, control systems in order to figure out what their parameters are. Now I'll give you a quick example with, this smaller little slinky. So let's just say we had it hanging And, instead of having access, we can't go in and just pull down on the end of the Slinky and watch it oscillate back and forth. Instead, we can only hold the top of it and move the top around. Now, in order to get the bottom part to move, we can either kind of oscillate the top of it. Or we can give it a quick tug. You can see at some points, the spring is going to start bumping into the top. So, one thing that we try to do in research is figure out what the optimal way to excite the top of the Slinky would be in order to figure out parameters such as damping or even the spring constant. And we do this for a lot more complex systems. but the basic idea is the same, that we're using spring-mass-damper systems, and we're trying to estimate parameters within the system. Now, for this demo we're going to keep it real simple and we are going to just pull on the end of the slinky. And, we're going to give it some sort of Energy input, since we're stretching out the spring and letting it go. Now there's no real way, also, to estimate with a ruler or apple or anything. We can't use any of that to estimate the damping, so we're going to use the tools of simulation to help us figure out what our damping coefficient is. So the next step of the demo is going to be using MatLab or Python whatever your choices of programming language. You're going to create a simulation of the spring mass damper system using the equation motion that we figured out here. Now once you do that, I've created this simulation You can create a visualzation if you'd like, of the spring. And also we're going to plot the position of the spring over time. And this will really help us to figure out what the damping parameter is. As well as check other paramenters that we've already derived. So once you have this system working You can give it a try, and I will let it play for a few seconds. You can see that our spring is oscillating up and down, and we have, over time, the position oscillating also, with our visualization. Now the key part of this is going to be testing out the simulation. And I'll start running it one more time [SOUND] and, at some point in the simulation, you're going to want to stretch your spring down and try to get, let it go so it becomes in phase with the simulation. Now if your parameters are pretty close, you should have the spring match up for a little while with your demo and Given that, you can watch how much your spring dampens out and see if it follows the trajectory that you've plotted. So, what I mean by that is if you take a look at your plot of position over time, you can see that there is a definite slope here. To the position. Now, what this indicates that we have putting damping into the system through this damping coefficient term. Now, changing what your value for B is will effect how quickly, your position over time decays. So, if you have a high damping coefficient, you'll see a response. Closer to this, and on the bottom as well, it will match up. So if you have a higher coefficiant, it will contract much quicker. All your energy will, all your energy will be lost, all right? If you have a small dampen coefficient, you won't see it level off quite as much, It will take it a long time for it to stop. Moving up and down. And this will really depend on what system you choose. Slinkies don't have too much damping so you're going to have a relatively low damping value. something like a rubber band or a hair tie may have more damping. And a good test would be to see what different damping coefficients are of different systems that you can find. In this case, we found, I matched it up. And you can see on the demo, it was okay. You're not going to get it perfect, as any engineered system, but you can get close with a simulation. So I found that the standing coefficient was about 0.007 Newton-seconds per meter. >> So, depending on your system, you can figure out if that is a reasonable term, if that is too high, too low, and you can also check out your spring constant and mass parameters as well. If your spring and mass are off, you might see that your simulation is either going up and down too fast, too slow And if your parameters are just right, it should stay pretty much in phase with your actual Slinky or, spring mast damper system. So, using this technique, you can try to match up simulation with a physical system to determine if your parameters are, are correct in estimating the physical system. Now as the last constituents g/ law that I hinted to in the introduction, we're going to take a look at a electrical system. Now I know we haven't talked about electrical systems in class yet, but for those who are familiar with electricity we're going to talk about Ohm's Law. Now, we're going to discuss electricity and [INAUDIBLE] laws about week seven or eight, so you can always refer back to this if you want to and try out this demo as well. So, ohm's law is given by the equation V equals I times R. V is our voltage. I is current and R is resistance. And to test this out we're going to create a simple circuit. We're going to hook a battery Directly to a resistor. This battery wil provide the volltage, across the resistor, which has a specified resistacne, and once we hook it up we're going to have current fowing through the circuit. Now, as you can see, this is a linear relationship, very much similar to the viscis damping and the spring constant system that we have found. And in a way, they are all sort of related. So to test this out, we need to measure both the voltage and current in the circuit. To do this, we're going to use two mufti-meters, mufti-meters are generally a good way to measure any sort of characteristics in a circuit, and it's pretty necessary because you can't just get a ruler to measure out distance or anything like that. So we have our circuits set up, and we'll take a look at the values, and on the left is our voltage reading and the right is the current reading. So, if we write down the values here we have 5.85 volts, and we have 1.26 milliamps. So given these two quantities, we can figure out what the resistance is of our resistor. So we'll just plug that into the equation. 5.85 volts equals 1.26 milliamps Times R, and we're going to divide 5.85 by 1.26 milliamps. Make sure you divide this my a thousand as well because we're going to use SI units, so you want volts and amps. And the result is going to be 4,640 ohms. Now generally we want to reduce this to a smaller number. So we say, 4.64 kilo ohms. So, how do we check whether this is right? Well, fortunately resistors have specificed resistances on themselves, there's a little colored bands on a resistor. you can look on the internet for how to read those. And, I read this off, and the specified one is 4.7 kilohms. And that's what the resistor's rated for. However, even in engineered systems you'll see that there's a difference between what we'll measure and what the actual parameter is rated for. And manufacturers take that into account. There's a tolerance that's, also displayed on the resister and this one is plus or minus 5%. So you can see that our measured value is well within that range and therefore, our system is pretty linear in the resistance. So now we've looked at three different constitutive laws. The spring versus force law, the viscous damping law, and ohms law with electricity. You can see that these are all pretty much linear, but they are linear because we are using engineered systems. if you have given systems around the house, you may see that the numbers are off a little bit, but should be fairly close. So now go ahead and try one of these out on your own systems or try them all out. We're going to have a link that you can submit videos of your demos on and you can try to either do the demo that we've shown here in this video or feel free to go around your house anywhere around and try to estimate parameters of various objects. See if things are linear. See if they might not match up with the equations that we've shown here. And feel free to submit those videos to us and we're looking forward to seeing what those are. If you need a copy of the steps to replicate this demonstration, A PDF's available in the lecture notes, and please go through and try this out, and we're looking forward to seeing how you guys do. Thanks. [BLANK_AUDIO]