Page 204 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 6.9 Modal Damping in Forced Vibration 191

leads to the eigenvalues and eigenvectors that describe the normal modes of the
system and the modal matrix P or P. If we let X = PY and premultiply Eq. (6.9-1)
by P^ as in Sec. 6.8, we obtain
P^MPY + P^CPY + P^KPY = P^F (6.9-3)
We have already shown that P^MP and P^KP are diagonal matrices. In general,
P^CP is not diagonal and the preceding equation is coupled by the damping
matrix.
If C is proportional to M or K, it is evident that P^CP becomes diagonal, in
which case we can say that the system has proportional damping. Equation (6.9-3)
is then completely uncoupled and its iih equation will have the form

y, + 2Ci(o,y, + (ojy, = M t) (6.9-4)
Thus, instead of N coupled equations, we would have N uncoupled equations
similar to that of a single-DOF system.

Rayleigh damping. Rayleigh introduced proportional damping in the
form
C = aM F ßK (6.9-5)
where a and ß are constants. The application of the weighted modal matrix P
here results in
P^CP = aP^MP + ß p K P
= a i F ßA {6.9-ey

where / is a unit matrix, and A is a diagonal matrix of the eigenvalues [see Eq.
(6.7-6)].

(o]
OJj
A = (6.9-7)

Thus, instead of Eq. (6.9-4), we obtain for the iih equation

y, + {a + l3o)j)y^ •+ wfy, = /'(i) (6.9-8)
and the modal damping can be defined by the equation
= a -t- ^iof (6.9-9)

can be shown that C = + /3/C" can also be diagonalized (see Probs. 6-29 and 6-30).