Page 216 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Chap. 6 Problems 203

6-25 Using the modal matrix P, reduce the system of Prob. 6-24 to one that is coupled only
by damping and solve by the Laplace transform method.
6-26 Consider the viscoelastically damped system of Fig. P6-26. The system differs from the
viscously damped system by the addition of the spring /cj, which introduces one more
coordinate, jcj, to the system. The equations of motion for the system in inertial
coordinates x and Xj are
nvc = -kx - c ( i - i | ) -t- F
0 = c ( i — i j ) — k^x^
Write the equation of motion in matrix form.

Figure P6-26.

6-27 Show, by comparing the viscoelastic system of Fig. P6-26 to the viscously damped
system, that the equivalent viscous damping and the equivalent stiffness are

/c -f (/cj + ^)(x7)
k... =
- ( ? r '

6-28 Verify the relationship of Eq. (6.6-7)
d>jK4> = 0 i ^ j
j
by applying it to Prob. 6-16.
6-29 Starting with the matrix equation

premultiply first by KM~' and, using the orthogonality relation = 0, show that
= 0
Repeat to show that
<t>J[KM-'f K4>^ = 0
for /i = 1, 2,..., n, where n is the number of degrees of freedom of the system.