200
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
Lumping a portion of the structure modeled as a rigid body is not so subjective as that for a framework. For instance, for a rigid deck in plane rotation
about a node at its mass center G, or for a concrète monopod in plane rotation
about a ftxed base point 0, ail of the mass is used. For such a model, the rigid
structural mass is assumed to be symmetric with respect to a vertical centerline.
With this symmetry assumption, the diagonal element will be the mass moment
of inertia with respect to the node point and ail of the associated products of
inertia terms due to antisymmetrical mass distribution disappear, leading to
zéro off-diagonal terms in the mass matrix. An example of a diagonal M for
the coupled motion between a rigid deck and its flexible supporting structure is
given later in this chapter.
A variation of the lumped mass method called the consistent mass theory
can also be used to calculate M. This theory, however, is usually quite tedious to
implement and is beyond the scope of the présent text. For further discussions,
the reader is referred to the following expositions: Clough and Penzien (1993),
who applied this theory to beams and frames; Chopra (2001), who illustrated
the method for plane frames; and Utku (1984), who based his rigorous analysis
on the principle that the sum of the kinetic energy for each discretized structural
element of a plane frame or truss is equal to the kinetic energy for the whole
structure. It is noted that the consistent mass matrix method leads to a banded,
symmetric matrix with some non-zero and some négative off-diagonal terms.
This writer has found that a judicious modeling of offshore structural supporting
framwork using the lumped mass method first discussed usually leads to quite
satisfactory results, whereas refinements achieved by employing the consistent
mass matrix to the same model lead to results for dynamic responses that are
nearly the same in many cases.
In applying the lumped mass methods, it is always very important to interpret the mass of ail submerged components as Virtual mass. It is this writer’s
expérience that to neglect the use of Virtual mass can lead to an error in a structurel fondamental frequency of 40 percent to 50 percent; and such an error can
lead to comparable errors in the structure’s dynamic responses.
Example Problem 8.3. Compute now the Virtual mass for Example Problem
8.1, shown in Figure 8.1. Here, nodes 1 and 2 are on the vertical centerline of the
structure, in line with the base point 0, and the System mass is symmetrically
distributed with respect to this centerline. Choose node 1 at the mass center
of the deck and its equipment, at the height
+ £2) from 0 at the sea floor.
Locate node 2 at the mass center of the horizontal members, at height ^2 fr°m
0. The masses lumped at nodes 1 and 2 are approximated as follows:
mi =m„ + 0.375 mvl
(8.9)
m2 =
+ 0.375 mvi + 0.375 mV2
(8.10)
In the last équations, mp is the total actual mass of the deck and its equipment
(not submerged), and mvh is the virtual mass of the horizontal members at
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
Lumping a portion of the structure modeled as a rigid body is not so subjective as that for a framework. For instance, for a rigid deck in plane rotation
about a node at its mass center G, or for a concrète monopod in plane rotation
about a ftxed base point 0, ail of the mass is used. For such a model, the rigid
structural mass is assumed to be symmetric with respect to a vertical centerline.
With this symmetry assumption, the diagonal element will be the mass moment
of inertia with respect to the node point and ail of the associated products of
inertia terms due to antisymmetrical mass distribution disappear, leading to
zéro off-diagonal terms in the mass matrix. An example of a diagonal M for
the coupled motion between a rigid deck and its flexible supporting structure is
given later in this chapter.
A variation of the lumped mass method called the consistent mass theory
can also be used to calculate M. This theory, however, is usually quite tedious to
implement and is beyond the scope of the présent text. For further discussions,
the reader is referred to the following expositions: Clough and Penzien (1993),
who applied this theory to beams and frames; Chopra (2001), who illustrated
the method for plane frames; and Utku (1984), who based his rigorous analysis
on the principle that the sum of the kinetic energy for each discretized structural
element of a plane frame or truss is equal to the kinetic energy for the whole
structure. It is noted that the consistent mass matrix method leads to a banded,
symmetric matrix with some non-zero and some négative off-diagonal terms.
This writer has found that a judicious modeling of offshore structural supporting
framwork using the lumped mass method first discussed usually leads to quite
satisfactory results, whereas refinements achieved by employing the consistent
mass matrix to the same model lead to results for dynamic responses that are
nearly the same in many cases.
In applying the lumped mass methods, it is always very important to interpret the mass of ail submerged components as Virtual mass. It is this writer’s
expérience that to neglect the use of Virtual mass can lead to an error in a structurel fondamental frequency of 40 percent to 50 percent; and such an error can
lead to comparable errors in the structure’s dynamic responses.
Example Problem 8.3. Compute now the Virtual mass for Example Problem
8.1, shown in Figure 8.1. Here, nodes 1 and 2 are on the vertical centerline of the
structure, in line with the base point 0, and the System mass is symmetrically
distributed with respect to this centerline. Choose node 1 at the mass center
of the deck and its equipment, at the height
+ £2) from 0 at the sea floor.
Locate node 2 at the mass center of the horizontal members, at height ^2 fr°m
0. The masses lumped at nodes 1 and 2 are approximated as follows:
mi =m„ + 0.375 mvl
(8.9)
m2 =
+ 0.375 mvi + 0.375 mV2
(8.10)
In the last équations, mp is the total actual mass of the deck and its equipment
(not submerged), and mvh is the virtual mass of the horizontal members at