BEAM RESPONSES
269
With the orthogonality conditions of équations (10.70) and (10.71), each term in
the sum of équation (10.87) vanishes except for m = n, in which case équation
(10.87) reduces to
ÿn(t) + w’ yn(t) - Pn(t)
(10.88)
in which
1
f
Pn(t) = — y Xn q(x, t)dx
(10.89)
(10.90)
Assume that the beam is initially at rest, or that v(x,0) — dv(x,t)/dt — 0.
Using équation (10.82), the initial rest position implies that t/n(0) = t/n(0) = 0.
The solution to équation (10.88) is in the form of the Duhamel intégral for C = 0,
or
îtn(i) = — / Pn(r) sinun(t - rjdr
Jü
(10.91)
This is verified by comparing équation (5.58) to équation (10.88) and then by
comparing their respective solutions given by équations (5.76) and (10.91).
These results are summarized. The transverse response v(i, t) for arbitrary
transverse unit loading q(x, t) of an undamped Bernoulli-Euler beam, subjected
to one of the six constraints of Table 10.1, is given by équation (10.82). Here
and wn are calculated as outlined in Section 10.1, and yn(t) is calculated
from équations (10.89)-(10.91).
Parametric Excitation
Consider the dynamic behavior of a vertical dredge pipe attached at the top
to a barge as shown in Figure 10.12a. The barge undergoes harmonie heave
motion in regular waves. This pipeline is modeled as the beam of Figure 10.12b
where the ends are simply supported to avoid adverse bending stresses at thés
points of fixity. The average longitudinal load on the pipeline is assumed in the
form
P = Pû + Pi cos dit
(10.92)
where Po is approximated as the sum of the ballast weight (in water) applied
at the lower end (x — 0) and one-half of the pipeline’s weight (in water). Note
that in reality the hinge at i = f carries ail of the pipeline’s weight. whereas the
hinge at x — 0 carries none of the pipeline’s weight. The approximation given
f°r Po becomes more accurate for increasingly high ratios of ballast weight to
pipeline weight.