Repeat Example but for three masses with the m’s =

⎡ ⎢ ⎢ ⎣ 2 − x 2 p 2 − 1 − 1 2 − x 2 p 2 − 1 − 1 2 − x 2 p 2 − 1 − 1 2 − x 2 p 2 ⎤ ⎥ ⎥ ⎦ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ y 1 y 2 y 3 y 4 ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = 13.10 The curvature of a slender column subject to an axial load P Fig. P13.10 can be modeled by d 2 y d x 2 + p 2 y = where p 2 = P E I where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis. This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give y i + 1 − 2y i + y i − 1 x 2 + p 2 y i = where i = a node located at a position along the rod’s inte- rior, and x = the spacing between nodes. This equation can be expressed as y i − 1 − 2 − x 2 p 2 y i + y i + 1 = Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equa- tions. For example, if the column is divided into five seg- ments i.e., four interior nodes, the result is An axially loaded wooden column has the following charac- teristics: E = 10 × 10 9 Pa, I = 1.25 × 10 ⫺5 m 4 , and L = 3 m. For the five-segment, four-node representation: a Implement the polynomial method with MATLAB to determine the eigenvalues for this system. b Use the MATLAB eig function to determine the eigen- values and eigenvectors. c Use the power method to determine the largest eigen- value and its corresponding eigenvector. 13.11 A system of two homogeneous linear ordinary differ- ential equations with constant coefficients can be written as d y 1 dt = − 5y 1 + 3y 2 , y 1 0 = 50 d y 2 dt = 100y 1 − 301y 2 , y 2 0 = 100 If you have taken a course in differential equations, you know that the solutions for such equations have the form y i = ce λ t where c and λ are constants to be determined. Substituting this solution and its derivative into the original equations converts the system into an eigenvalue problem. The result- ing eigenvalues and eigenvectors can then be used to derive the general solution to the differential equations. For exam- ple, for the two-equation case, the general solution can be written in terms of vectors as {y} = c 1 {v 1 }e λ 1 t + c 2 {v 2 }e λ 2 t where {v i } = the eigenvector corresponding to the i th eigen- value λ i and the c’s are unknown coefficients that can be determined with the initial conditions. a Convert the system into an eigenvalue problem. b Use MATLAB to solve for the eigenvalues and eigen- vectors. c Employ the results of b and the initial conditions to determine the general solution. d Develop a MATLAB plot of the solution for t = 0 to 1. 13.12 Water flows between the North American Great Lakes as depicted in Fig. P13.12. Based on mass balances, the following differential equations can be written for the concentrations in each of the lakes for a pollutant that decays with first-order kinetics: a b x y y x P⬘ P 0, 0 L, 0 P⬘ M P FIGURE P13.10 a A slender rod. b A freebody diagram of a rod.