NATURAL FREQUENCES OF LINEAR SYSTEMS
107
If the structure consists of vertical beams and a deck of weight of md9 at x =
as for the jackup platform of Figure 2.17, then the decrease in the gravitational
potentiel energy of the deck as the horizontal leg displacement increases is
Vg = -mgg Ah
(5.19)
where Ah is the vertical drop in height of the deck due to curvature of the elastic
beams (legs). The beam element of arc length ds shown in Figure 2.17b can be
approximated using a binomial expansion, or
ds — (dx2 + dv2Ÿ^2 ~ dx 1
1
2
(5.20)
Thus the drop in height of this arc element is
,
,
1 ( dv
ds — dx = - ( ——
2 \ dx
2
dx
(5.21)
from which the total drop of
is calculated by integrating over the length of
the beam, or
(5.22)
With the latter resuit and équation (5.19), the decrease in the potential energy
of the deck mass due to the latéral leg motion is thus
(5.23)
Consider the kinetic energy for the beam and deck éléments. For a submerged beam element of length dx with a virtual mass per unit length of m, the
kinetic energy is m(dv/dt')2dx/2. (If the element is not submerged, replace m by
mo, its actual mass per unit length.) The kinetic energy for a fully submerged
beam such as in Figure 2.16a is thus
(5.24)
For N beams or legs only partially submerged, such as in Figure 2.17, the resuit
is
2
N
dx + — I rriQ
2 Jd
(5.25)
where d is the water depth. For this jackup platform there is an additional
kinetic energy term due to the movement of the deck mass. If the rotational