28
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
The free stream horizontal flow velocity u = u(t) is in line with the cylinder s
translational motion v - v(t). As shown on its free body sketch, the restraint
and damping forces per unit length that oppose the cylinder motion are kv and
ci), respectively. The Virtual mass per unit length, m, is deduced from équation
(2.9). The fluid loading per unit length is given by équation (2.14), modified so
that the drag force is based on the relative velocity (u - û) between the fluid
and the cylinder. When Newton’s second law, équation (2.1), is applied to this
cylinder, the équation of motion becomes
Çm0 + C.4 p?r^-^ v + cv + kv = CdP-^\u - ï>|(u - û) + Cm Pn~^'‘ (2.21)
The term involving Cm results from fluid motion only, such as wave action,
where û is the absolute accélération of the fluid.
Equation (2.21) is nonlinear as a conséquence of the drag force term. Berge
and Penzien (1974) linearized this équation for small motion, or
C'D = Cd |u — — constant
(2.22)
With this assumption, équation (2.21) becomes
D2
D
D2
= CDp—u + Cm
(2-23)
Equation (2.23) clearly shows that the damping of a moving cylinder is increased
due to the fluid drag force, a force that in general overwhelms the internai
structural damping c. The conclusion holds true for its nonlinear counterpart
also, équation (2.21).
Thus, to solve équation (2.21) or (2.23) for the structural displacement
v — v(t), one needs to know four structural parameters: mg, D, c, and fc;
the fluid density p; the free stream flow field u = u(t); and the three empirical
constants C4, Cd and Cm ■ In subséquent examples, k and c will be estimated for
particular cases, and the dependency of the latter three empirical constants on
offshore waves will be discussed in greater detail. Other environmental factors
affecting the motion of offshore structures are now quantified.
Buoyancy and Gravity
It is well known that a solid object can be lifted much more easily when
in water than in air. This is because the water pressure exerts an upward or
buoyant force on the submerged solid. The Greek mathematician Archimedes
(287-212 B.C.) stated this principle in précisé terms:
.4 solid body partially submerged in a fluid is buoyed up by a force
mkg equal to the weight of the fluid displaced. (mb is the mass of the
fluid displaced.)
STRUCTURE-ENVIRONMENTAL FORCE INTERACTIONS
The free stream horizontal flow velocity u = u(t) is in line with the cylinder s
translational motion v - v(t). As shown on its free body sketch, the restraint
and damping forces per unit length that oppose the cylinder motion are kv and
ci), respectively. The Virtual mass per unit length, m, is deduced from équation
(2.9). The fluid loading per unit length is given by équation (2.14), modified so
that the drag force is based on the relative velocity (u - û) between the fluid
and the cylinder. When Newton’s second law, équation (2.1), is applied to this
cylinder, the équation of motion becomes
Çm0 + C.4 p?r^-^ v + cv + kv = CdP-^\u - ï>|(u - û) + Cm Pn~^'‘ (2.21)
The term involving Cm results from fluid motion only, such as wave action,
where û is the absolute accélération of the fluid.
Equation (2.21) is nonlinear as a conséquence of the drag force term. Berge
and Penzien (1974) linearized this équation for small motion, or
C'D = Cd |u — — constant
(2.22)
With this assumption, équation (2.21) becomes
D2
D
D2
= CDp—u + Cm
(2-23)
Equation (2.23) clearly shows that the damping of a moving cylinder is increased
due to the fluid drag force, a force that in general overwhelms the internai
structural damping c. The conclusion holds true for its nonlinear counterpart
also, équation (2.21).
Thus, to solve équation (2.21) or (2.23) for the structural displacement
v — v(t), one needs to know four structural parameters: mg, D, c, and fc;
the fluid density p; the free stream flow field u = u(t); and the three empirical
constants C4, Cd and Cm ■ In subséquent examples, k and c will be estimated for
particular cases, and the dependency of the latter three empirical constants on
offshore waves will be discussed in greater detail. Other environmental factors
affecting the motion of offshore structures are now quantified.
Buoyancy and Gravity
It is well known that a solid object can be lifted much more easily when
in water than in air. This is because the water pressure exerts an upward or
buoyant force on the submerged solid. The Greek mathematician Archimedes
(287-212 B.C.) stated this principle in précisé terms:
.4 solid body partially submerged in a fluid is buoyed up by a force
mkg equal to the weight of the fluid displaced. (mb is the mass of the
fluid displaced.)