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Math 2250 Lab 14 Name/Unid: 1. Consider the multiple mass-spring system shown in the figure below. We neglect the curvature so that each mass is pulled in opposite directions by the springs that connect its two neighbors. Recall that in an ordinary single mass-spring system that the restorative force is proportional to the change in the length of the spring so that mx00 = −kx Imagine, now that there are two masses, m1 and m2 , connected by a single spring with spring constant k. We then have the system m1 x001 = −k(x1 − x2 ) m2 x002 = −k(x2 − x1 ) m1 x1 m5 x2 x5 x4 m4 x3 m3 m2 (a) Suppose that all 5 springs have the same spring constant, k. Use the above information to construct a system of differential equations for the system of the form Mx00 = Kx where x1 x2 x= x3 x4 x5 Specify the matrices M and K. Page 2 (b) Let mi = 1 for i = 1, ..., 5. This reduces the above equation to x00 = Kx. Let y = x0 . Convert the 5 × 5 system from part (a) to a 10 × 10 system for the vector x1 .. . x x z= = 5 y y1 . .. y5 This will have the form z0 = Az where A is a 10 × 10 matrix. Page 3 (c) Suppose we have an 2n × 2n matrix A= B C D E where B, C, D, E are all n × n matrices. Then, if DE = ED, we have the result that det(A) = det(BE − CD). Use this fact to show that if λ is an eigenvalue of the K in part (a), then ω = is an eigenvalue of the matrix A from part (b). Page 4 √ λ (d) Find the eigenvalues of K. What are all the possible frequencies that the system can oscillate at? Without actually computing the eigenvectors, write down the general solution of the system in terms of linear combination of functions of t times the eigenvectors. You may use any technology you want to find the eigenvalues. Page 5 (e) Find a non-trivial equilibrium solution to the system in (a). How does this relate to the eigenvector corresponding to λ = 0? Give a physical interpretation of this result. Page 6 2. Consider the network of tanks shown in the figure below. r1,in = 3 [L/m] c1,in = 20(1 + sin(2π ∗ t)) x1 (t) V1 = 20 r3,1 = 1 [L/m] r2,1 = 2 [L/m] x3 (t) V3 = 10 r2,3 = 1 [L/m] x2 (t) V2 = 10 r2,out = 3 [L/m] (a) Let x1 x = x2 , x3 c1,in and cin = c2,in c3,in denote the vector of the amount of dissolved salt in each tank and the vector of salt concentrations going into each tank, respectively. Derive the system of differential equations of the form x0 (t) = Ax(t) + c(t) (Use the next page if you need more space.) Page 7 Page 8 (b) Find the eigenvalues and eigenvectors of A. Page 9 (c) Write down the complementary solution xc that satisfies the homogeneous system. Page 10 (d) Determine the particular solution xp . Page 11 (e) Write the full general solution to the system in (a). Page 12 3. An n × n matrix is called diagonalizable if there exists an invertible matrix S, such that A = SΛS−1 where Λ is a diagonal matrix (that is Λij = 0 for all i 6= j). (a) Show that for any integer k Ak = SΛk S−1 Page 13 (b) Show that if λ1 λ2 Λ= .. . λn then Λ = k λk1 λk2 Page 14 ... λkn (c) Recall that the exponential of a matrix is defined as ∞ X 1 k A . e = k! k=0 A Use this along with the results from parts (a) and (b) to show that e λ1 eλ2 A −1 e =S .. . Page 15 e λn S (d) The above shows that we can compute the matrix exponential if we know the eigenvalues of the matrix, A, and the matrix S. It turns out that S is a matrix with the eigenvectors of A as column vectors. If λi is an eigenvalue with corresponding eigenvector vi then S = v 1 · · · vn , λ1 Λ= λ2 . ... λn That is, the eigenvectors in S are in the same order as the eigenvalues in Λ. Use this to compute by hand eA when A= Page 16 1 2 2 1