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SolSec 4_9 - Problems and Solutions for Section 4.9(4.80...

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Problems and Solutions for Section 4.9 (4.80 through 4.90) 4.80 Consider the mass matrix M = 10 ! 1 ! 1 1 " # $ % & and calculate M -1 , M -1/2 , and the Cholesky factor of M . Show that LL T = M M ! 1/2 M ! 1/2 = I M 1/2 M 1/2 = M Solution: Given M = 10 ! 1 ! 1 1 " # $ % & The matrix, P , of eigenvectors is P = ! 0.1091 ! 0.9940 ! 0.9940 0.1091 " # $ % & The eigenvalues of M are ! 1 = 0.8902 ! 2 = 10.1098 From Equation M ! 1 = P diag 1 " 1 , 1 " 2 # $ % & ( P T , M ! 1 = 0.1111 0.1111 0.1111 1.1111 # $ % & ( From Equation M ! 1/2 = Vdiag " 1 ! 1/2 , " 2 ! 1/2 # $ % & V T M ! 1/2 = 0.3234 0.0808 0.0808 1.0510 # $ % & ( The following Mathcad session computes the Cholesky decomposition.
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4.81 Consider the matrix and vector A = 1 ! " ! " " # $ % & ( b = 10 10 # $ % & ( use a code to solve Ax = b for ε = 0.1, 0.01, 0.001, 10 -6 , and 1. Solution: The equation is 1 ! " ! " " # $ % & ( x = 10 10 # $ % & ( The following Mathcad session illustrates the effect of ε on the solution, a entire integer difference. Note that no solution exists for the case ε = 1. So the solution to this problem is very sensitive, and ill conditioned, because of the inverse.
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4.82 Calculate the natural frequencies and mode shapes of the system of Example 4.8.3. Use the undamped equation and the form given by equation (4.161). Solution: The following MATLAB program will calculate the natural frequencies and mode shapes for Example 4.8.3 using Equation (4.161).
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