[SOUND] Hi, and welcome back. In today's module, we're going to get into some of the more complicated aspects of fatigue. We'll be looking at randomly varying stresses. So, the learning outcomes for today's module, are to understand the difference between a fluctuating stress, and a randomly varying stress. To understand the principles behind Miner's Rule and to understand the point at which failure will occur with D, the accumulated damage. So, lets get started. We've talked a lot about failure theories in this class and in the past few modules we have been looking at fatigue. So, we already talked about Fully Reversed stresses and how you can directly use the SN diagram. Then we talked about Fluctuating stresses, and we looked at how to calculate the mid range stress and the alternating stress and use Goodman's line to determine life and finite versus infinite life. And today, we're going to get into varying fluctuating stresses, which are really common in design. So, if we look at the top of the screen here, we can see a fluctuating stress. And you can see it's constant with time. So while this stress is going from 40 to -20, it repeats that exact cycle 40 to -20 for quite some time. Then, if we look at the bottom of the screen, we can see a randomly fluctuating or a randomly varying stress. And here, we have a couple cycles going from 40 to -20. But then the cycle suddenly spikes up to 90, down to -60. And then it comes in and starts to go somewhere in the 70 to -40 range. And then it starts to be quite small. So, it's varying. If we think about mechanical design, just in a car. So if we're thinking about a car, if a car is driving in a straight line, at a constant speed, some of its components, say, the drive shaft, will be feeling these fluctuating stresses that are constant over time. But as soon as the car starts to accelerate or brake, or change speeds, or turn, the forces on all of the components in the car are going to change. And so, it's much more common in mechanical engineering design that we end up with these randomly fluctuating stresses, depending on operational modes. And it's important to be able to learn how to quantify this. So, for varying fluctuating stresses, or randomly varying stresses, you'll see these referred into many different manners. What we're going to do is we're going to think of the total life of the component. And we're going to look at each stress cycle and see how much life of the component is taken away due to that stress cycle. A great analogy of this is any type of video game. So if you played a video game, let's say we're playing a video game with a bumble bee and it starts out with a total life at the top of the screen, you get these life bars and you're full. And then say your bumble bee video game character has to fly a really long distance. And so, its life gets reduced. And then maybe it has to fight, I don't know, a spider, and its life gets reduced further. You can think of a mechanical engineering component in the same exact manner. So now, let's pretend the life bar at the top of the screen is, for our component, which is feeling the stresses to the left of the screen. In that case, if we look at the first stress cycle it's going to see, which is right here. So, these stress cycles, it sees two of these cycles. And we can say each one of these cycles takes away one life box. So we're going to lose these. And then it goes into this heavy, the biggest stress cycle it will see. And that stress cycle takes away a significant portion of the life. So we lose this because it only repeats once. And then it goes into these other cycles which are not quite as small as the first cycle and not as quite as big as the second. It repeats twice, so we're going to lose one and a half lives, and then another one and a half lives. And then finally, a certain portion of life, it goes into the smallest stress cycle, which takes away a small portion of the life. And so, if we look at the component totally, it only has so much life remaining. So, this analogy, where you start to look at each cycle in terms of life. The fundamental principle behind this is called Miner's rule, or the linear cumulative damage rule. And the rule is, D, which stands for damage. So D stands for damage, is equal to the sum of lowercase ni, divided by uppercase Ni. Lowercase n is the number of cycles at a stress level sigma i. So how many times are you seeing that stress? And upper case N is the number of cycles that would cause failure at that stress level i. So basically, they're looking to see, it's a ratio. How many cycles have you experienced? And how many cycles would cause failure? What's the ratio of that, that causes the damage? So here's kind of a mini-example of how to apply Miner's rule. Let's say we have a component, and this component initially sees stress cycles that go from -60 ksi up to 20 ksi, and those are repeated 5 times. You already know, you already learned to use the Goodman equation to figure out an equivalent life. And then use SN equations to calculate the life of these stress cycles. So how much life, if our component was just going to continue to see these cycles, would it have? The answer is, it could cycle at -60 ksi to 20 ksi for about 10 to the 4th cycles, or 10,000 cycles, right? In this case, since the component only sees these cycles 5 times, our lowercase n is 5. And our uppercase N is 10 to the fourth. So then let's say our component then starts to see stress cycles from -30 ksi to 10 ksi repeated 3 times. So in that case, our lower case n is 3. It's seeing three of 3 cycles. And if we can calculated the life of the component if it just saw these stresses, it's saying 5x10 to the 3. So our upper case N is 5x10 to the 3. So then what we can do is go ahead and calculate the damage. And in this case, the damage is 5 divided by 10 to the 4 + 3 divided by 5x10 to the 3. And that would be the damage of the part. And the idea is, when your D equals one, or when your damage is equal to one, your failure has occurred. And that makes sense, right? Because if we thought about say, we were going to have this, instead of being repeated 5 times, it was being repeated 10 to the fourth times. Then our damage would be equal n, which would be 10 to the fourth, divided by N=10 to the 4th, which is 1, right? So that makes sense. Okay, so now that we've introduced Miner's rule, let's look at a more complicated example. So in this case, this example we have a varying fluctuating stress shown to the left, and it's found at the critical location of a component. So we don't need to worry about applying stress concentrations in here. We're going to assume everything is captured in this graph. So this is the maximum stresses that this component is feeling. It says the material is steel, it gives us a fully adjusted endurance limit of 30 ksi, an ultimate strength of 120 ksi, and an f of 0.82. And what it's asking us is to determine, what is the accumulative damage of this part? So what is D? Essentially they want to know, what is D? And then what is the life of the part in hours if this stress pattern continues to repeat for the remainder of the part's life? So you can see this stress pattern is 10 seconds of what the part sees. So, we're assuming it will continue to see this pattern every 10 seconds throughout its life. So, I'm going to ask you to go ahead and try to work through this example on your own. In the next module we'll continue to work out this example. And that's it for today. I'll see you next time. [SOUND]