Homework answers / question archive / Modeling and Analysis of Mechanical Systems Problem Set 4 You must show all of your work or you will not receive credit

Modeling and Analysis of Mechanical Systems Problem Set 4 You must show all of your work or you will not receive credit

Mechanical Engineering

Share With

---

Modeling and Analysis of Mechanical Systems

Problem Set 4

You must show all of your work or you will not receive credit.

Problem 1. The quadratic f = x 2 +4xy+2y 2 has a saddle point at the origin, despite the fact that its coefficients are positive. Write f as a difference of two squares.

Problem 2. Decide between a minimum, maximum, or saddle point for the following functions:

a) F = −1+4(e x −x)−5x siny+6y 2 at the point x = y = 0.

(b) F = (x 2 −2x) cos y, with stationary point at x = 1, y = π.

Problem 3. The quadratic f(x1, x2) = 3(x1 + 2x2) 2 + 4x 2 2 is positive. Find its matrix A, factor it into LDLT , and connect the entries in D and L to the numbers 3,2,4 in f . That is, where do these numbers appear in D and L.

Problem 4. If A has independent columns, then A TA is square, symmetric, and invertible. Rewrite x TA TAx to show why it is positive except when x = 0. Then A TA is positive definite.

Problem 5. Show from the eigenvalues that if A is positive definite, so is A 2 and so is A −1 . Problem 6. The ellipse u 2 +4v 2 = 1 corresponds to A = 1 0 0 4 . Write the eigenvalues and eigenvectors, and sketch the ellipse.

Problem 7. From the pivots, eigenvalues, and eigenvectors of A = 5 4 4 5 , write A as R TR in three ways: (L √ D)(√ DLT ), (Q √ Λ)(√ ΛQ T ), and (Q √ ΛQ T )(Q √ ΛQ T ).

Problem 8. Find the minimum value (and the minimizing x) for (a) P(x) = 1 2 (x 2 1 +x 2 2 )−x1b1 −x2b2 (b) P(x) = 1 2 (x 2 1 +2x1x2 +2x 2 2 )−x1 +x2. 1

Problem 9. What equations determine the minimizing x for (a) P = 1 2 x TAx−x T b (b) P = 1 2 x TA TAx−x TA T b (c) E = kAx−bk 2 ?

Problem 10. Suppose u1,...,un and v1,..., vn are orthonormal bases for R n . Construct the matrix A that transforms each v j into uj to give Av1 = u1,...,Avn = un.

Problem 11. Find UΣV T if A has orthogonal columns w1,...,wn of lengths σ1,...,σn.

Problem 12. Find the SVD and the pseudo inverse VΣ +U T of B = 0 1 0 1 0 0

Problem 13. In the system shown below with three springs, two forces, and two displacements, write out the equations e = Ax, y = Ce, and A T y = f . For unit forces and spring constants, what are the displacements? 2

Problem 14. For the system shown to the right, find the corresponding equations e = Ax, y = Ce, and A T y = f . Solve the last equation for y (in terms of f 0 s). Why can we find y before we find x in this case?

Problem 15. For the same system as Problem 14., find K = A TCA and A −1 and K −1 . If the forces f1, f2, f3 are all positive, acting in the same direction, how do you know that the displacements x1, x2, x3 are also positive?  

Problem 16. Write down the incidence matrices A1 and A2 for the following graphs (A1 for the left graph and A2 for the right graph): For which right sides does A1x = b have a solution? Which vectors are in the nullspace of A T 1 ?

Problem 17. Consider an R by R network in the plane, with nodes at the N = R 2 points with integer coordinates between 1 and R. (a) If horizontal and vertical edges make it into a network of unit squares, find the number of edges. Show that the approximate ratio of m to N is 2 to 1. (b) If the network also includes the diagonals of slope +1 in each square, show that the approximate ratio of m to N for this triangular mesh is 3 to 1.

Problem 18. Show that the matrix M = C −1 A A T 0 = 1 1 1 0 is neither positive definite nor negative definite by finding its pivots and eigenvalues. Does x1 x2 1 1 1 0 x1 x2 = x 2 1 +2x1x2, have a minimum, maximum, or saddle point at x1 = x2 = 0?

Problem 19. Given a network and its incidence matrices (ungrounded and grounded): (a) Find A TCA following the notes for lecture 13, page 7 at the bottom (use c1, c2, c3 for the diagonal entries of C in this case. 4 (b) Find A TCA from “column-row” multiplications, where the first column of A T and the first row of A give: ? ? −1 1 0 ? ? c1 = −1 1 0 = ? ? c1 −c1 0 −c1 c1 0 0 0 0 ? ?. Add up the five products of this kind (one for each edge). 5