Q:

6.1 (a) From the determinant method, the polynomial equation for the natural frequencies of a three-DOF system is as follows: where .

Calculate the value of the lowest natural frequency (hint: 0 < ? < .5) and its corresponding mode shape. Normalize so that the largest value of the mode shape is +1 .0. Be accurate to two or more significant figures in your final answers.

(b)(c)(d) Use the determinant method to determine all three natural frequencies of the systems whose mass and stiffness matrices are listed below. Be sure to list the frequency solutions in their proper order.

(e) What is the mode shape for a single-DOF system? Hint: Such a mode shape would have only a single numerical entry.

Q:

6.2 (a) Calculate the first mode shape for part (b) above.

(b) Calculate the first mode shape for part (c) above.

(c) Calculate the first mode shape for part (d) above.

Q:

6.3 Use matrix iteration to calculate the first natural frequency and first mode shape for the structural systems whose mass and stiffness matrices are as follows

Q:

6.4(a) Use matrix iteration to determine the first eigenvalue and eigenvector when the three-DOF dynamic matrix is, by rows [1 1 1; 2 6 6; 2 6 14].

(b) Use matrix iteration to calculate the second eigenvalue and eigenvector when the diagonal mass matrix entries are [\1/2 1 1\]. Hint: It is suggested that your first...