Okay? And then, when we do this reduction of a second order ODE to two first order ODEs, for the time discretized problem, okay, we get the following form. We get the following form. We get y at n plus 1, okay, equals A, which is now our amplification matrix, y at n plus L at n. Okay? All right, and let me tell you just once more, that Y at n plus 1, is d at n plus 1, v at n plus 1. All right? This is of course dn, vn. And A here is a 2 by 2 amplification matrix. Okay? It plays the same role as our scalar amplification factor for our parabolic problem, okay? And the definition of A, as well as the definition of this two vector Ln, reflects the reflects the Newmark algorithms right, with the gammas and betas and everything. Okay? We can work out all these details, but it's just tedious detail, which we are not going to truly use. Okay? All right, so then this is our time discretized equation, now written in, in the form of two first order ODEs instead of one second order ODE. Okay? I'm going to give you a summary of the stability results right now. Okay, stability. Okay? So, if 2 beta is greater than or equal to gamma is greater than or equal to half, we have unconditional stability. Okay? If on the other hand, gamma being greater than or equal to half, beta lies between 0 and gamma over 2. These conditions together give us conditional stability. Okay? Now remember, we are talking about a single degree of freedom modal equation, okay? Where, because of the fact that we have a second order ODE, what we are solving for are dn plus 1 and vn plus 1, right, both of those modal coefficients. The conditional stability holds when omega h, right, which is the frequency, corresponding to that particular mode, right? Remember, as before we're suppressing the modes, okay? Actually, even back here, we are already suppressing. Mode, number, index, all right, L. Okay? So that's going on here as well. So when, even though I've just written omega h here, it's really omega hL for each L, right? We need to consider this for each L, okay. So the stability condition also requires that omega h delta t, should be lesser than or equal to a quantity that I'm going to denote as omega critical, okay? Where omega critical. Equals the following. Ch, and again, this is Ch sub l really, but for every mode, okay? C h times gamma minus half, plus gamma over 2 minus beta, plus c h gamma minus half, sorry, c h squared gamma minus half, the whole square. All of that to the power one half. The whole thing divided by gamma over 2 minus beta, okay? That's the critical frequency, okay? What we see here is that there is the effect of damping. Okay? And, what we also observe is that the effect of damping, right? And, and we have damping when c h is, greater than 0. The effect of damping is to increase the critical frequency, okay? So we have the undamped critical frequency. Let's say omega critical u for undamped, okay? This is got by setting c h equal to 0, okay? And it is just gamma over 2 minus beta to the power minus half, okay? Right? I just want to point out that this undamped critical frequency is a lower bound to omega critical, all right? So what we're seeing is that the undamped critical frequency is lesser than or equal to the actual critical frequency, when you have some damping, okay? All right. Okay? So what we see as well is that omega h delta t, which needs to be less than the, than the critical frequency, is, it satisfies this sort of a condition. Okay? What I mean by saying this is that actually, let, sorry, let me not write this line. This is, is sort of attempt to state a condition. Let me not write that, that equation, it, it can be misinterpreted. Instead let me say this. The undamped critical frequency is a more stringent condition. On omega h delta t, right? It's really a condition on delta t. So what we are saying is that instead of saying that it has to be less than the actual critical frequency, right? Instead of using the condition that I have at the bottom of this slide, right? If instead of that, we were to say that, well, omega h delta t has to be less than this quantity, okay? Then we are actually imposing a more stringent condition upon our algorithm, our time integration algorithm, okay? Okay, with this in hand, I'm just going to list out properties of some sort of canonical, almost classical, members of this Newmark family, okay? And, I'm going to do this part in a table, where I'm going to list the method here. I will say what type it is. And by type I mean is it implicit or explicit? I will list here beta, gamma, let me see, what else do I need to list here, right? I will list here the critical frequency for stability for the undamped case. And finally, I will also write here the order of accuracy. Okay? So, the methods we are going to consider are the following, the first one we will consider is the, what is sometimes called the Trapezoidal Rule. We consider four methods, okay. I'll write them out first, trapezoidal rule, think linear acceleration. We have the average acceleration. These are all names of methods. And finally we have the central difference method. And, when I say central difference and trapezoidal method, trapezoidal rule, note that they, they will not be the same as what you may be familiar with from first-order ODEs. Simply because we are using terminology here that has been established for second-order ODEs, okay? All right, all of these methods are implicit except for the, except for the central difference method. Okay? Now, stability. They're all for, they all use gamma equals half, okay? Now, the trapezoidal rule uses beta equals one quarter. And because this combination of beta and gamma makes it unconditionally stable, there is no question of what the critical frequency is for stability, right? It's unconditionally stable, all right? So there's nothing to say there. Linear acceleration uses one-sixth. And what happens here is that the the critical frequency is 2 root 3, okay? Average acceleration uses beta equals 112, gamma equals half. And if I remember the undamped critical frequency is square root of 6. The central difference method finally uses beta equals 0. And the undamped, actu, critical frequency here is 2, okay? For order of accuracy, all of these are second order. Okay? One caveat though is that the explicit you get a truly explicit algorithm only for M and C being diagonal. All right? It's just a summary of some of members of the family. As you can imagine, because we are talking of a, of integration algorithms of second order ODEs, the numbers of this family are, are, are a few more, right? It's a fairly large family. Okay, we can afford to stop this segment here.