8
Multi-Degree of Freedom
Linear Structures
James F. Wilson
The first approximation to determining the motion of a structure in the offshore
environment is to model that structure as a single degree of freedom System
in which the motion is described by a single coordinate. This was done in
previous chapters. For instance, for an exploratory drilling rig rigidly fixed to
the mat at the sea floor, the sway motion was computed in terms of a single
displacement coordinate v = v(t) at the top of the rig; and for a rigid monopod
gravity platform on a compliant subsea soil foundation, the rotational motion
was computed in terms of a single rotational coordinate 0 — 0(t).
In a refined dynamic analysis, several independent coordinates are used to
describe structural motion, hence the term multi-degree of freedom System. For
the monopod gravity platform, for instance, if the rigid deck had flexible connections to the rigid leg, then the rotation of the deck could be described by a
coordinate
and the rotation of the legs by another coordinate f)?. This dynamic model is a two degree of freedom System described by two ordinary, coupied differential équations of motion involving these two coordinates. Coupling
occurs since one motion affects the other through the flexible deck-leg interface.
More than two independent coordinates could be defined if one needed to account for the flexibility of the legs and the deck, and these coordinates could
include horizontal displacement coordinates.
In this chapter, differential équations for multi-degree of freedom structural
models are derived using both the Newtonian and the Lagrangian approach and
solved using the popular normal mode method. These classical théories were
freely drawn and condensed from the expositions of Chopra (2001), Clough
and Penzien (1993), and Utku (1984), in which some changes in structural
nomenclature were made to avoid redundancy with the coramon symbols of fluid
mechanics. The basic assumptions used to formulate the structural models are:
(1) the number of independent coordinates N chosen to describe the motion is
equal to the number of degrees of freedom; (2) the restoring forces are linear
functions of the chosen coordinates (linear structures); and (3) the damping is
linear-viscous. Numerical examples illustrating these models are given in the
following chapters.
195
Multi-Degree of Freedom
Linear Structures
James F. Wilson
The first approximation to determining the motion of a structure in the offshore
environment is to model that structure as a single degree of freedom System
in which the motion is described by a single coordinate. This was done in
previous chapters. For instance, for an exploratory drilling rig rigidly fixed to
the mat at the sea floor, the sway motion was computed in terms of a single
displacement coordinate v = v(t) at the top of the rig; and for a rigid monopod
gravity platform on a compliant subsea soil foundation, the rotational motion
was computed in terms of a single rotational coordinate 0 — 0(t).
In a refined dynamic analysis, several independent coordinates are used to
describe structural motion, hence the term multi-degree of freedom System. For
the monopod gravity platform, for instance, if the rigid deck had flexible connections to the rigid leg, then the rotation of the deck could be described by a
coordinate
and the rotation of the legs by another coordinate f)?. This dynamic model is a two degree of freedom System described by two ordinary, coupied differential équations of motion involving these two coordinates. Coupling
occurs since one motion affects the other through the flexible deck-leg interface.
More than two independent coordinates could be defined if one needed to account for the flexibility of the legs and the deck, and these coordinates could
include horizontal displacement coordinates.
In this chapter, differential équations for multi-degree of freedom structural
models are derived using both the Newtonian and the Lagrangian approach and
solved using the popular normal mode method. These classical théories were
freely drawn and condensed from the expositions of Chopra (2001), Clough
and Penzien (1993), and Utku (1984), in which some changes in structural
nomenclature were made to avoid redundancy with the coramon symbols of fluid
mechanics. The basic assumptions used to formulate the structural models are:
(1) the number of independent coordinates N chosen to describe the motion is
equal to the number of degrees of freedom; (2) the restoring forces are linear
functions of the chosen coordinates (linear structures); and (3) the damping is
linear-viscous. Numerical examples illustrating these models are given in the
following chapters.
195