SolSec 4.9 - Problems and Solutions for Section 4.9 (4.72...

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Problems and Solutions for Section 4.9 (4.72 through 4.82) 4.72 Consider the mass matrix M = 10 1 11 and calculate M -1 , M -1/2 , and the Cholesky factor of M . Show that LL M MM I M T = = = −− 12 // Solution: Given M = 10 1 The matrix, P , of eigenvectors is P = 0 1091 0 9940 0 9940 0 1091 .. The eigenvalues of M are λ 1 2 0 8902 10 1098 = = . . From Equation MP P M T = = 1 1 0 1111 0 1111 0 1111 1 1111 diag λλ ,, From Equation M Vdiag V M T = [] = 1 2 0 3234 0 0808 0 0808 1 0510 / / , The following Mathcad session computes the Cholesky decomposition.
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4.73 Consider the matrix and vector Ab 11 0 10 = ε εε use a code to solve Ax = b for = 0.1, 0.01, 0.001, 10 -6 , and 1. Solution: The equation is 0 10 = x The following Mathcad session illustrates the effect of e on the solution, a entire integer difference
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4.74 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 (4.151).
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SolSec 4.9 - Problems and Solutions for Section 4.9 (4.72...

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