196
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
8.1 EQUATIONS OF MOTION: GENERAL FORM
The general set of équations representing structural motion investigated in this
chapter are defined by the following matrix form:
MU C’U KC-p
(8.1)
Equation (8.1) represents a finite set of N ordinary, linear differential équations
in N independent coordinates (U,€2>--- ,-U )• 1!1 which the coefficient
matrices M.K, and C are constant. The single and double overdots of a coordinate represent the velocity and accélération, respectively, of that coordinate.
That is, ( ) is the operator d/dt. The notation is that each upper case, boldfaced
letter represents an (N x N) matrix, and that each lower case, boldfaced letter
(£. p) represents a (IxN) matrix or column vector of time-dependent éléments.
\Vhen N = 1, équation (8.1) reduces to a single degree of freedom structural
model, équation (2.43), and the respective terms of these two équations hâve
an analogous meaning. That is, the respective values of
p.M, K,and C are:
the coordinate vector, the loading vector, and the mass, stiffness, and damping
matrices. Each of these terms is now discussed in general and evaluated for
simple représentations of offshore structures, for N — 2 and 3.
The Coordinate Vector, £
The first step in modeling an offshore structure for dynamic analysis is to
carefully define a set of N independent coordinates U- f = 1,2,..., N, that
contain the dominant features of the structural motion. These coordinates are
the éléments of the column vector
£ =
....... g-]t
(8-2)
in which the superscript T dénotés transpose (the interchange of the row éléments shown to its defined column array). This vector can be composed of a
mixture of displacement coordinates, designated in applications as Vi, V2, • • • ,
and rotational coordinates, designated as /'!.'<■.
. Each chosen coordinate
describes the motion of a node point on the structure, such as the mass center
of a structural element or a junction point on a frame. Each chosen coordinate
must then be tested as follows for independence. Freeze the motion of (N - 1)
node points and then check whether the lone remaining node can hâve motion
when loaded. If and only if motion occurs at that lone node is its associated
coordinate independent. Then repeat this mental test for each of the remaining
nodes, one-by-one, to check for their independence.
Example Problem 8.1. To illustrate the choice of independent coordinates,
consider the simple représentation of a jacket template structure shown in Figure 8.1a, a two-bay, four-legged tower fixed at the sea floor and in plane motion.
This tower, which is symmetric about its vertical centerline, is a scaled-down
version of the fîve-bay configuration discussed by Mansour and Millman (1974).
Assume that. as the whole tower sways side-to-side with wave loading, the deck
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
8.1 EQUATIONS OF MOTION: GENERAL FORM
The general set of équations representing structural motion investigated in this
chapter are defined by the following matrix form:
MU C’U KC-p
(8.1)
Equation (8.1) represents a finite set of N ordinary, linear differential équations
in N independent coordinates (U,€2>--- ,-U )• 1!1 which the coefficient
matrices M.K, and C are constant. The single and double overdots of a coordinate represent the velocity and accélération, respectively, of that coordinate.
That is, ( ) is the operator d/dt. The notation is that each upper case, boldfaced
letter represents an (N x N) matrix, and that each lower case, boldfaced letter
(£. p) represents a (IxN) matrix or column vector of time-dependent éléments.
\Vhen N = 1, équation (8.1) reduces to a single degree of freedom structural
model, équation (2.43), and the respective terms of these two équations hâve
an analogous meaning. That is, the respective values of
p.M, K,and C are:
the coordinate vector, the loading vector, and the mass, stiffness, and damping
matrices. Each of these terms is now discussed in general and evaluated for
simple représentations of offshore structures, for N — 2 and 3.
The Coordinate Vector, £
The first step in modeling an offshore structure for dynamic analysis is to
carefully define a set of N independent coordinates U- f = 1,2,..., N, that
contain the dominant features of the structural motion. These coordinates are
the éléments of the column vector
£ =
....... g-]t
(8-2)
in which the superscript T dénotés transpose (the interchange of the row éléments shown to its defined column array). This vector can be composed of a
mixture of displacement coordinates, designated in applications as Vi, V2, • • • ,
and rotational coordinates, designated as /'!.'<■.
. Each chosen coordinate
describes the motion of a node point on the structure, such as the mass center
of a structural element or a junction point on a frame. Each chosen coordinate
must then be tested as follows for independence. Freeze the motion of (N - 1)
node points and then check whether the lone remaining node can hâve motion
when loaded. If and only if motion occurs at that lone node is its associated
coordinate independent. Then repeat this mental test for each of the remaining
nodes, one-by-one, to check for their independence.
Example Problem 8.1. To illustrate the choice of independent coordinates,
consider the simple représentation of a jacket template structure shown in Figure 8.1a, a two-bay, four-legged tower fixed at the sea floor and in plane motion.
This tower, which is symmetric about its vertical centerline, is a scaled-down
version of the fîve-bay configuration discussed by Mansour and Millman (1974).
Assume that. as the whole tower sways side-to-side with wave loading, the deck