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Point) Consider the following truss system.AIl bars are either vertical, horizontal, or at 60" from horizontal.Enter the elongation matrix (A = BT): (in the fo...

Question

Point) Consider the following truss system.AIl bars are either vertical, horizontal, or at 60" from horizontal.Enter the elongation matrix (A = BT): (in the form "node horiz" , "node vert" , "node 2 horiz" etc_2+1/sqrt2Ilsqrt21/sqrt21+1/sqrt2(Remember that webwork uses radians for computations_This truss system should be stable: The matrix is square, which is good. However; you should be able to verify that A has a pivot in each column of its LU decomposition:

point) Consider the following truss system. AIl bars are either vertical, horizontal, or at 60" from horizontal. Enter the elongation matrix (A = BT): (in the form "node horiz" , "node vert" , "node 2 horiz" etc_ 2+1/sqrt2 Ilsqrt2 1/sqrt2 1+1/sqrt2 (Remember that webwork uses radians for computations_ This truss system should be stable: The matrix is square, which is good. However; you should be able to verify that A has a pivot in each column of its LU decomposition: This means that it has no nullspace_



Answers

What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?

And this question uh question 13.116 It is exactly the same as question 13.115 Which we have to fight if the column of the bill 12 time 40 is adequate for supporting the fourth P equal to uh 15 kip. But this time there is another allowable stress for the pending stories which is 15 Ks. I wish it is. It will be another restriction for this question. And first of all, we have to fight the ko ratio of X. S and Y, Y axis. And in this case I have found that and Kl over uh for X axis is greater. So we will use the value of care or up will be 111.2. And next we have to find the um okay, uh see for a 36 still which is 126.1. And we will see that 100 11.2 is less than 100 26.1. So uh this column is the intermediate column. And then we have to use this equation to find allowable stress and I found that it is 11.51 case I and next you have to find um the allowable uh Now we have to fight, we have to use this equation. The lie about um stories is equal to P or a plus EMC or I. But every day why the equation by the liable is raised. You have that P over a libel stressed plus EMC or I a reliable stress. But um this one is the component of the bending moment. And the question give us the value of the allowable stress for the bending moment which is 15. So for this term we have to use 15 case I fought the allowable swiss and for this one we can use um 11.51 case I for this. Now if we calculate that you see equal to four point oh four and m is 1 80 P is of course is 15 and a is 11.51 and I is um is this one 56.3 and we'll have that. It is equal 2.9 431 which is less than one. So um if we calculate the ratio of um strangers and the allowable stress, you see that it is 15 divided by 14.7, which is um That's right and a little stressed here is the 1.51 and you get that is equal to point oh 89 which is less than 0.15 So um this column of w 12 time 50 is adequate for supporting the first p.

This question that is a 30 36 steel of W. 12 50 column. And there is an apply force of 15 G. I. P. Apply to the to the column as shown in the figure in the textbook. And uh the question one has to find that if the column is adequate to support that load. Um First we have to find A K. A. L. Or R ratio of the column in both X and Y. Y exits. So first start with exit and exit K. Is too. And how is 24 time 12. And our is 5418 And that means K. L. Over our X. Is 100 element too. And then Y. Y. Acted. The K. Is 27 and ry is 1.96 That makes that's why you is 102.8. Okay we can see that. Okay. All over our supply is less than care about subjects. So we have to use okay all sub eggs to um to find a lot of stress. But before that we have to shake um the control K. All over us, Fc. For the a third status still which can be found from this equation. And we'll have that okay over us, Fc. Is 126.1. And we'll see that um the value of kor some X. Is less than gail over our. So see so this column is the intermediate column. So we have to use this equation to find the allowable stress last three A. Or eight. And hey to the power three eight we're where A. Is K. L. O. R. Water bike are see now from that we'll have that the allowable stresses 11.512 years. I So uh we have a lot of those stories so that we can calculate ah maximum strays from the following force and moment which is P divided by A plus EMC, or I. Peace 15 divided by 14.7 plus M. Is 100 and 80. Um 4.4 divided by 56 43. And we have that the maximum stress is 13 point 94 K. S. I. Now we can see that the maximum stress is greater than the liable stress so that this column U. 12 I'm 50 is not adequate for supporting the first and moment. P. And M.

Hi it is exercise. We have here the following transformations represented in the plane. And we need to find the Eigen vectors and Eigen values using only the geometric meaning. So in this case we are talking about reflection about the y axis. That means that here we have the X Y coordinate, you have the x coordinate. An idea is that you have any point in the plane. And what you're going to do is reflected with respect to the y axis. So in this case you're going to pass this point here to this new point over here. So you're reflecting basically this vector, Let's see P to a new vector over here. That's going to be the multiplication of these matrix A. To the point B. Okay, so you're reflecting with respect to Y. Now, what is the meaning of Eigen space? And I can values and Eigen vectors basically is the day in space is going to be the points that are going to be in variant under this transformation. You can observe that if you pick a point in the y axis then the reflection will return to the same point. So basically here, a a queue here, Q and a Q are going to be the same point. The same here here here. So no matter which point in the y axis you choose, you're going to obtain the same point. So that's the finding in space, Dragon space of a. It's defined by the Y axis basically. Now we need to find, what would you define and I get back to basically what you're doing is finding and vases for the dragon space. So you can see that i in fact a basis for the Y axis will be the victor 01. And there's a nagging spectrum of this transformation. Part of the organ basis, basically the basis for the Eigen space will be a spun by this picture. Now this is the second picture. What can we say about the against pagan values? Well, we know that if we take these this matrix and we lie to the back turbie, we should obtain here some scalar and the vector. Right? But in this case This lambda here is the cost to one because this victory turned the same victory. So the lambda associated to the second factor in particular islam that equals two. The next transformation that we have here is the rotation about the origin. This case again we have here the white access here we have the X axis. And the idea is that again you have some point in the space some vector. And what you're going to do is rotate this vector 180°. So basically 180°. It returned you a vector in the opposite direction. And also you can observe just by the form of the matrix. Okay that you want to resolve this uh algebra likely that all the vectors are going to satisfy the agon equation digging equation. Remember that is equals to the metrics. The transformation in this case times a venture returns you to some constant let's say new. I'm here the same vector. So this is the wagon equation. These vectors are the Eigen vectors and new is going to be the Eigen value basically. Here for all the vectors in the space we have that relation because what we're doing here, it's just uh taking like the negative victor, right? You can observe just from from from the from the matrix. What we're doing is We have a victory be then after applying a TV. What we have here is -5. So from here you can observe what's going to be the value of the Eigen value and this is equal to -1. And technically all R two is the dragon space. Okay took for any vector in the space this is going to satisfy the egg an equation. Okay, that means well, if you want to find some basis, basically you have here the trivial basis for the egg in space. That is economical because the egg and space. Easy plane. So 10 And 01. We only have here trivial solutions. Okay, the next one is a dilation factor here. I want to mention some important thing. And is that no matter which value here you have if you have some skill is killing off the identity matrix because this you can regret it sk and the identity matrix. And you you observe the same in the previous example because we have here just the identity minus the identity matrix. So we have an scaling of the identity matrix and no matter which is the constant that is going to multiply the identity matrix here we are. Um this system has a solution. Could you broadly, of course a victor equals two K. This K. Here of the dilation factor and the same vector. And this is satisfied for all the vectors in the plane in particular. So the dragon space is going to be the reels. The planes are so the dragon space of this made transformation is going to be the reels. The wait, that means that as in the previous example, the basis for this iron space or the Eigen vectors, it's going to be the canonical basis for the space. Now, this is like the other report but we're interested in the um geometric meaning. So basically what we have here is this is a plane. And this election fracture. What is going to do is if you have some point here then because it's a dilation, you're going to enlarge the vector by a factor of K. Of course this value of K can be negative. So in that case you're going to put to the opposite direction. But basically all these vectors are going to be just risk scale and that satisfied equation of the mhm Eigen values. So for any any any vector here in the plane, you will have an dilation and risk killing of the spectrum. So no matter which I've actually choose, then you have that the whole um plane, it's going to satisfy this equation. So the asian space is going to be art. Finally we have here this year in the Y direction. So the sheer is geometrically looks like this. And for that, I'm going to use a unit square to illustrate what happened with this year. So let's consider here the unit here like that. Okay, here, Yeah, yeah, so there's the unit square, there's a vector 10. And this vector over here is a vector 01. Okay, so what I'm doing here is using this union square to illustrate what is the geometric meaning of the sheer in the right direction. Basically what you're going to do is taking these vectors, all these vectors that are pointing to the Y direction. You're going to translate them to this direction by a sheer. Okay, that's here. It's going to put all the vectors to one K. And here you do the same. You have a better over here. So you can see you're sharing basically the unit square, That's not the only one. You have the other in the negative direction. So this is the .1. Okay. And you're going to move old in that direction. Okay, one. And you have this. Okay, so will you see like kind of mess rolling here. But the point is to see what is left in variant in this United Square and you can observe that what is left embarrassment is the vectors that are pointing to the Y direction. I'm sorry here, I got a mistake here. This is not the matrix, there's a matrix for to share in the right direction. So basically no matter which point you have in the Y axis under this transformation, you're going to be unaffected just in the Y axis. So that means that the egg in space for this transformation is generated by this pan of the vector 01. That there is a vector that point in the uh direction, if this is going to be the icon vector then what happened if we apply the sheer to this vector? So basically you're going to multiply this matrix one, K 1. And because this vector is left in variant, it returns you the same vector just by the geometric meaning that we have here. You can see that this vector in particular never change. Uh the Eigen value is going to be what


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