A . l In t r o d u c t i o n
In this appendix we develop explicit and exact functions for the total potential energy o f prismatic bodies o f rectangular cross-sections o f arbitrary aspect ratio. These simple geometric shapes can be thought o f as first approximations to ship-like floating structures. The expressions presented here can then be useful in verifying some basic hydrostatic properties o f interest to us (see section 3.7.1), as well as allowing, under certain assumptions, the development o f equations o f motion valid for large-amplitude regimes.
We consider a body of breadth E , depth XE, length b, and mass m. Such a body
has three planes o f symmetry: two vertical (longitudinal and transversal) and one horizontal. We assume that the centre o f gravity G is located along the line defined by the intersection o f the two vertical planes o f symmetry, and at a
distance a from the baseline, see figure A. 1.
Figure A.2 defines basic coordinates and geometrical parameters. Vertical displacements (heave motions) are defined by the distance between G and the water surface (positive upwards), and denoted by v . Angular displacements (roll motions) are defined as the angle between the body and the water surface (zero
for the upright position, and positive clockwise), and denoted by 6.
The gravitational potential energy function V (v,6) is given by the sum o f body
Appendix A: Potential Energy Functions 195
6/2
^ —
Figure A. 1 - Main dimensions of prismatic body o f rectangular cross-section
A . l
We adopt a zero potential energy for the body when its centre o f gravity is at the water surface, and we assume that the water displaced by the body forms an infinitely thin film on the water surface, so that the individual potential energies
can be given by V j,^(v) = mgv and = pgbAvg, where p is the water
density, g is the gravitational acceleration, A is the submerged area (o f the cross-
section), and Vg is the depth o f the buoyancy centre as shown in figure A.2.
Expressions for can be more easily determined if we consider three different
cases as shown in figure A. 3. Cases 1, 2, and 3 are defined in terms of the relative position between the body and the water surface: in Case 1 the water surface crosses both vertical sides o f the body, in Case 2 one vertical side o f the body is crossed by the water surface, along with the bottom o f the body, and finally in Case 3 one vertical side is crossed along with the top o f the body. If we
Appendix A : Potential Energy Functions 196
(-)v
Figure A.2 - Basic coordinates and geometric parameters for the floating square cross-section body
CASE
CASE
CASE
Appendix A: Potential Energy Functions 197
V, = 0 - sin é) A.2
Vj = ^ { X c o s 0-\- sin ê) A 3
V = V -
2 ; Then the following relations define Cases 1 to 3:
e \
a ~ X — \cosO A 4
Case 1; -v , <v^<Vj A.5
Case 2: A. 6
Case 3: -v, A. 7
Note that we assume so that values o f larger than modulus
correspond to situations in which the body is either completely emerged (v^ > 0) or completely submerged (v^ < 0). These situations will not concern us here.
A .2 Ex p r e s s i o n s f o r p o t e n t i a l e n e r g y
Expressions for V (v,6) in terms of its two independent variables can be
quite long. We therefore recall that the total potential energy is:
V (v,6) = mgv-\-pgbA(v,e)Vg(v,e) A.8
It will then suffice to present expressions for the submerged cross-sectional area
A (v,6) and for the position o f the centre o f buoyancy v^(v,6). To keep expressions even shorter we choose to introduce a few auxiliary parameters, and
Appendix A: Potential Energy Functions 198
geometrical parameter for the following calculations is given by D, the water surface elevation measured along the body vertical centre line (see figure A.2):
D = a —
cosO A.9
Case 1
The expressions for A and for Case 1 are given by:
A = ED A.IO
y 2 24D COS 6
A.11
Case 2
The expressions for A and for Case 2 are given by:
A = 1 2 tan 6 tanO+D A.12 V o = - DcosO-\— sinO 2 j A.13 Case 3
The submerged cross-sectional a re a ^ can be given by:
A = ÀE^-A. A. 14
Appendix A: Potential Energy Functions 199
, \ ( E (A £-£))^
A = - — ^
y 2 tan 0 j
2
The depth of the centre of buoyancy can be given by:
V g = d c o s 0 A.16
Where the auxiliary parameter d and further auxiliary parameters are given by:
d = D - Ô ^ ^ - Ô y ^ t a n O A . 1 7 S.. = A.19 l = X E - - \ E t a n 6 + X E - D \ A.20 3 \ 2 a . 2 1 2 3 l 2 tone A 3 Re s u l t s f o r a h o m o g e n e o u s b o d y o f s q u a r e c r o s s-s e c t i o n
In order to get some feel for the shape o f potential energy surfaces as we vary both heave and roll displacements we consider the specific case o f a homogeneous body o f square cross-section. Here, of course, the centre of gravity G will remain fixed at the geometric centre of the body. We, however,
allow the average density of the body, p, to vary, and we then inspect the
resulting hydrostatic properties, such as stable and unstable floating positions, as well as potential energy surfaces. By doing so we will at the same time be checking on the feasibility of one o f our assumptions (see equation 3.27) regarding the development of the SIR equations.