6 A system shown in figure 3 has m 4 k1 100 k2 100sqrt2 and

6. A system shown in figure 3 has m =4; k1 =100 ; k2 =100*sqrt(2) and c = sqrt(3); The corresponding springs and damper are installed as shown in figure 3. Find natural frequencies ?n1, ?n2, damping ratios ?1, ?2 and mode shapes u1 and u2 (Hint: u(.) is an eigenvector of M^(-1)K ) Suppose the damping coefficient c is increased to a certain value so that one of the modes becomes critically damped. Find the corresponding value of c.

Solution

Now, IN this case the displacement parallel to the dampner would be considered
Forces parallel top dampner, assuming displacement x along the dampner
k1x/cos(90-63.434) + k2x/cos(90 + 45 - 63.434) + mx\" + cx = 0
now
fro critical damping
c^2 = 4mk
c^2 = 4*m*[k1/cos(90-63.434) + k2/cos(90 + 45 - 63.434)] = 94.5761