SINGLE DEGREE OF FREEDOM STRUCTURES
19
damping force of the general form f(v). When équation (2.1) is applied in the
v direction to this free body sketch, the resulting équation for surge motion
becomes
mv + f(i>) + q(v) =pi(t)
(2.2)
In the first term of équation (2.2), the virtual mass m for surge motion is about
15 percent higher than the actual ship’s mass. Quantitative values for the
remaining terms of this équation are discussed later in this chapter.
Other examples of single degree of freedom offshore structures include the
pure, plane rotational motion of buoys such as shown in Figures 1.8 and of
gravity platforms rocking in the vertical plane, such as depicted in Figure 2.2.
With ail out of plane motion and translational motion supressed, the angular
displacement of these structures, modeled as rigid bodies of virtual mass m, the
angular displacement is uniquely described by the single coordinate 0 = 0(t).
Suppose that such a structure rotâtes about a fixed point 0 in the plane. Let
EAlo dénoté the sum of ail external moments in the plane of motion, acting on
m, positive in the positive direction of 0. These moments, which are due to the
four types of external forces discussed above, are ail expressed with respect to
the same fixed point 0. In this case, the équation of motion has the general
form
52 Mo = Jo0
(2.3)
in which 0 is the absolute angular accélération of the rigid body and Jq is the
virtual mass moment of inertia of this body with respect to the reference axis
through point 0 and perpendicular to the plane of motion. The value of Jq is
defined as
Jo = [ r2dm
(2.4)
Jm
where r is the distance from that reference axis to the virtual mass element
dm, and the intégration is over the whole rigid body. In applications it is often
convenient to express Jg in terms of Jg, or the value of Jq when the point 0
coïncides with the mass center G. The connection is through the parallel axis
theorem, or
Jq — Jq +
(2.5)
where ho is the distance between 0 and G. Values of J g for a variety of solids
of uniform density are listed in most elementary texts on rigid body dynamics.
For a relatively rigid structure composed of such shapes, the structure’s total Jq
value can be estimated by calculating Jq for each elementary component using
équation (2.5) and then superimposing the results.
19
damping force of the general form f(v). When équation (2.1) is applied in the
v direction to this free body sketch, the resulting équation for surge motion
becomes
mv + f(i>) + q(v) =pi(t)
(2.2)
In the first term of équation (2.2), the virtual mass m for surge motion is about
15 percent higher than the actual ship’s mass. Quantitative values for the
remaining terms of this équation are discussed later in this chapter.
Other examples of single degree of freedom offshore structures include the
pure, plane rotational motion of buoys such as shown in Figures 1.8 and of
gravity platforms rocking in the vertical plane, such as depicted in Figure 2.2.
With ail out of plane motion and translational motion supressed, the angular
displacement of these structures, modeled as rigid bodies of virtual mass m, the
angular displacement is uniquely described by the single coordinate 0 = 0(t).
Suppose that such a structure rotâtes about a fixed point 0 in the plane. Let
EAlo dénoté the sum of ail external moments in the plane of motion, acting on
m, positive in the positive direction of 0. These moments, which are due to the
four types of external forces discussed above, are ail expressed with respect to
the same fixed point 0. In this case, the équation of motion has the general
form
52 Mo = Jo0
(2.3)
in which 0 is the absolute angular accélération of the rigid body and Jq is the
virtual mass moment of inertia of this body with respect to the reference axis
through point 0 and perpendicular to the plane of motion. The value of Jq is
defined as
Jo = [ r2dm
(2.4)
Jm
where r is the distance from that reference axis to the virtual mass element
dm, and the intégration is over the whole rigid body. In applications it is often
convenient to express Jg in terms of Jg, or the value of Jq when the point 0
coïncides with the mass center G. The connection is through the parallel axis
theorem, or
Jq — Jq +
(2.5)
where ho is the distance between 0 and G. Values of J g for a variety of solids
of uniform density are listed in most elementary texts on rigid body dynamics.
For a relatively rigid structure composed of such shapes, the structure’s total Jq
value can be estimated by calculating Jq for each elementary component using
équation (2.5) and then superimposing the results.