Page 276 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 8 Problems 263

8-3 For the system in Sec. 8.1, the eigenvector for the first mode was determined by the
Gauss elimination method. Complete the problem of finding the second and third
eigenvectors.
8-4 For Prob. 8-1, rewrite the characteristic determinant as
1 1.5 -0.5 0]
-A 1 -r -1 2 - 1 = 0
1 0 -1 1J
by dividing the first equation by 2. (See App. C.4.) Note that the new determinant is
now not symmetric and that the sum of the diagonal, or trace, is 4.5, which is the sum
oh the eigenvalues. Determine the eigenvectors from the cofactors as in App. C.4.
8-5 In the method of cofactors, App. C.4, the cofactors of the horizontal row, and not of
the column, must be used. Explain why.
8-6 Write the equations of motion for the 3-DOF system shown in Fig. P8-6, in terms of
the stiffness matrix. By letting 2 ^ ^ ^2 = = k, the roots
of the characteristic equation obtained from program POLY are Aj =0.198, A2 =
1.555, and A3 = 3.247. Using these results, calculate the eigenvectors by the method of
Gauss elimination and check them against the eigenvectors obtained by the computer.

^ —V\AA^— —W W — ^2 —W\A/— m j

"^3 Figure P8.6.
8-7 Repeat Prob. 8-6 starting with the flexibility equation.
8-8 Draw a few other diagrams of systems equivalent to Fig. P8-6, and determine the
eigenvalues and eigenvectors for /c, and m, assigned by your instructor.
8-9 Determine the equation of motion for the system shown in Fig. P8-9 and show that its
characteristic equation is (for equal k^ and m^)

'
A^ - 9A^ -h 25A^ - 21A + 3 = 0
Solve for the eigenvalues and eigenvectors using the POLY program.

- w w -

g-WWW- —WVA/— ^ V W \^ m3 ^ v w v —
i
"^4 Figure P8.9.

8-10 Using the eigenvalues of Prob. 8-9, demonstrate the Gauss elimination method.
8-11 In Example 5.3-2, if the automobile wheel mass (m,) for the two front wheels and the
same for the two rear wheels) and tire stiffness (/cq for the two front tires and the same
for the two rear tires) are included, the 4-DOF equation of motion in matrix form