240
APPLICATIONS OF MULTI-DEGREE OF FREEDOM ANALYSIS
2. Calculate each component pfc(t) of the loading vector p(t). Select linear
wave theory. Use Morison’s équation and diffraction theory where appropriate
to détermine the components of loading.
3. Calculate each component function G(pk,aj) of the load transfer vector
G(p,w).
4. Calculate the undamped frequencies
and the normalized mode shape
matrix X. The consecutive columns of X are modal vectors xj,X2,... ,x\. The
fcth modal vector has components
X2k, • • • , 2 N/.
5. Define the generalized loading vector in modal coordinates as q(t) = q.
Then calculate the corresponding transfer functions G(qk, ! for each component
qk of q where
9fc = Xfc p(t)
(9-60)
Thus the generalized load transfer function for the fcth mode is
G(qk, w) = x£ G(p, w)
(9.61)
Note that G(p,w) was calculated in step 3.
6. With équations (8.95) and (9.60), the uncoupled équations of motion
become
ÿk +
= qk
(9.62)
Calculate F/fc(u>), the harmonie response function for the fcth mode, by substituting the following quantities into the last équation:
qk(fy =
yk(t) = Hk^)e^
(9.63)
The results, including the modulus, are
Hju’l = (u2 k -w2 + 2Xfcu>fcw)-1
(9.64)
1^)1 = H - J)2 + (2Cfc^)2]-112
(9.65)
7. Assume that p(t) is a stationary ergodic process. It follows that q(t)
will be stationary and ergodic also since the components of p(t) and q(t) are
related by the linear transformation (9.60). Using the analysis in Chapter 7 for
the single degree of freedom System in the form of équation (9.62), it follows
that the spectral density of yk is
S(yk,^ = \Hk^)\2S(,qk,^
(9-66)
Here S(qk,uj) is the spectral density of the generalized force component qk8. Assume linear wave theory. From Example Problem 7.2, deduce the
follow ing analogous relationship between the spectral density of the fcth load,
Sfpfc.u?), the wave height spectrum Sn(w), and the transfer function G(pk>u)'
S(.pk, w) — |G(pit, w)|2 S^(w)
(9-67)