10. 1. 2. 3. 4.
5. 6.
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SCHWAB, AL, MEIJAARD, JP & PAPADO- POULOS, JM, Benchmark results
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the
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linearised equations of motion of an uncontrolled bicycle, Proc of ACMD’04, Seoul, Korea, 2004
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APPENDIX ONE – DEFINITION OF THE HULL SHAPE USED AS A BASIS FOR FIGURE 1
The underwater cross-sections are defined to be semi- ellipses, and waterline beam and draught vary as 1 - 2x/lWLn, where x is measured from amidships and lWL is the waterline length. n is chosen to give the required
(mYvd) (mzmYpd) (mxmYrd) (mzmLvd) (IxxLpd) (mxmNvd) (IzxNpd) Yv Yp (Yrmu0)
(IxzLrd) v 0.242/√Cf = log10(
Re.Cf)
where Re is the Reynold’s Number based on waterline length.
APPENDIX TWO – DEVELOPMENT OF THE MATHEMATICAL MODEL
Substitution of equations 5-7 into equations 1-3 leads to the following, written in matrix form:
v p
.
(IzzNrd) r 0
0
. .
=
prismatic coefficient Cp. Beam, draught, wetted area and the position of the metacentre can be calculated from Cp or n, lWL and the specified beam/draught ratio. Values of residuary resistance coefficient are taken from [9] for three values of Cp, 0.55, 0.59 and 0.63, three values of beam/draught ratio, 2.25, 3.0 and 3.75, and three values of displacement volume/lWL
3, 0.001, 0.0015 and 0.002.
Interpolation, and some modest extrapolation, is used for other displacement ratios. For each hull length, that value of Cp is chosen that gives the lowest total resistance. It is 0.55 except when lWL is well below the optimum value. The friction coefficient Cf is calculated from the Schonherr formula which was used in deriving the results in [9]:
Yδ
Lv Lp (Lr + mzmu0) p + gm(zmetzm) + Lδ δ Nv Np (Nrmxmu0) r
Nδ
We also have = p. If we denote the four matrices above as M, D, E and F respectively, then by pre-multiplying by the inverse of M, we can write all of the above in the form:
. v
p = M-1D r.
. .
or . 0 1 x = Ax + bδ . where x is the 4-element vector xT = [ v p r ]. 0 v
M-1E p + M-1F δ r
0 0
B-10
©2007: Royal Institution of Naval Architects
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