EQUATIONS OF MOTION: GENERAL FORM
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node 2. The subscripts fl and v2 for the other mass terms dénoté Virtual mass
for portions of the legs and nonhorizontal members (or supporting structure)
surrounding nodes 1 and 2, respectively. For simplicity, assume that ail members
below the deck are fully submerged. The coefficient 0.375 is the proportion of
the structure’s mass that surrounds a particular node point, deduced from a
somewhat similar cantilevered structure using energy considérations, Example
Problem 5-4- The remaining mass of the whole System can be considered lumped
at the fixed node point 0, and is of no conséquence. The virtual masses for each
node are approximated by
n^vh — (actual mass of horizontal members at node 2) + Ca pVh
mvi = (actual mass of legs and X-members in tf) + Ca pVi
mv2 — (actual mass of legs and X-members in £2) + CApV'2
Recall that the quantities in added mass terms are: the added mass coefficient
Ca (which, lacking any experimental data for this geometry, can be approximated as unity); the mass density of water p; and the volumes of water Vh, V|
and V2 displaced by the respective structural components: the submerged horizontal members, the submerged members surrounding node 1, and the submerged members surrounding node 2. The mass matrix for this example is
thus
M = diag(mi, m2)
(8.11)
where uq and m2 are given by équations (8.9) and (8.10).
The Stiffness Matrix, K
The stiffness matrix for an N degree of freedom structure is a symmetric
array of N x N éléments in the following form:
A’h
ki2
k2l
^22
kiïi
kfw
kjN
k2N
kNN
(8-12)
The constants kij, where i,j = 1,2,... ,N, are the static stiffness influence
coefficients, which are generated from the actual structure after each node point
and its associated coordinate is identified, but before any attempt is made to
lump the mass. In this section, a displacement
means a generalized statit
displacement of node i: either a translation of node i or a rotation about node
t. Also, a force qsl means a generalized static force at node i. a simple forte
causing a displacement of node i along the line of action of that force, or a
moment causing a rotation about node i.
201
node 2. The subscripts fl and v2 for the other mass terms dénoté Virtual mass
for portions of the legs and nonhorizontal members (or supporting structure)
surrounding nodes 1 and 2, respectively. For simplicity, assume that ail members
below the deck are fully submerged. The coefficient 0.375 is the proportion of
the structure’s mass that surrounds a particular node point, deduced from a
somewhat similar cantilevered structure using energy considérations, Example
Problem 5-4- The remaining mass of the whole System can be considered lumped
at the fixed node point 0, and is of no conséquence. The virtual masses for each
node are approximated by
n^vh — (actual mass of horizontal members at node 2) + Ca pVh
mvi = (actual mass of legs and X-members in tf) + Ca pVi
mv2 — (actual mass of legs and X-members in £2) + CApV'2
Recall that the quantities in added mass terms are: the added mass coefficient
Ca (which, lacking any experimental data for this geometry, can be approximated as unity); the mass density of water p; and the volumes of water Vh, V|
and V2 displaced by the respective structural components: the submerged horizontal members, the submerged members surrounding node 1, and the submerged members surrounding node 2. The mass matrix for this example is
thus
M = diag(mi, m2)
(8.11)
where uq and m2 are given by équations (8.9) and (8.10).
The Stiffness Matrix, K
The stiffness matrix for an N degree of freedom structure is a symmetric
array of N x N éléments in the following form:
A’h
ki2
k2l
^22
kiïi
kfw
kjN
k2N
kNN
(8-12)
The constants kij, where i,j = 1,2,... ,N, are the static stiffness influence
coefficients, which are generated from the actual structure after each node point
and its associated coordinate is identified, but before any attempt is made to
lump the mass. In this section, a displacement
means a generalized statit
displacement of node i: either a translation of node i or a rotation about node
t. Also, a force qsl means a generalized static force at node i. a simple forte
causing a displacement of node i along the line of action of that force, or a
moment causing a rotation about node i.