The linear differential equations governing the motion of a ship in a sinusoidal wave disturbance are derived from the principles of rigid body dynamics. Stability criteria are obtained when the linear differential equations have (a) constant coefficients and (b) periodic time-dependent coefficients that are needed to describe the ship in a following sinusoidal wave. The frequency of encounter between ship and wave is not necessarily zero.
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References
1.
MandelP. ‘Ship manoeuvring and control’, Principles of naval architecture1967, chapter 8 (Soc. Nav. Archit. Mar. Engrs).
2.
DavidsonK. ‘A note on the steering of ships in following seas’, 8th Int. Congr. Applied Mechanics, London, 1948.
3.
WeinblumG. ‘On the directional stability of ships in calm water and in a regular seaway’, 1st U. S. Congr. Applied Mechanics, Chicago, 1951.
4.
SutherlandW.KroukovskyKorvin B. ‘Some notes on directional stability and control of ships in rough seas’, Stevens Institute of Technology, No. 91, 1948.
5.
WeinblumG.St DenisM. ‘On the motions of ships at sea’, Trans. Soc. nav. Archit. mar. Engrs195058, 184.
6.
RydillL. J. ‘A linear theory for the steered motion of ships in waves’, Trans. R. Instn nav. Archit.1959101, 81.
7.
LewisE. V.NumataE. ‘Ship motions in oblique seas’, Trans. Soc. nav. Archit. mar. Engrs196068, 510.
8.
WahabR.SwaanW. A. ‘Coursekeeping and broaching of ships in following seas’, Int. Shipbldg Prog.196411, 293.
9.
TasaiF. ‘On the swaying, yawing and rolling motions of ships in oblique waves’, selected papers from J. Soc. nav. Archit. Japan19683, 92.
10.
AbkowitzM. A. ‘The linearized equations of motion for the pitching and heaving of ships’, Proc. Symp. Behaviour of Ships in a Seaway, Netherlands Ship Model Basin, Wageningen, 1957, 178.
11.
VassilopoulosL.MandelP. ‘A new appraisal of strip theory’, Fifth Symp. Naval Hydrodynamics, Bergen, 1965, 253.
12.
KriloffA. ‘A general theory of the oscillations of a ship on waves’. Trans. Instn nav. Archit.189840, 135.
13.
GrimO. ‘Surging motion and broaching tendencies in a severe irregular sea’, Stevens Institute of Technology, No. 929, 1962.
14.
AbkowitzM. A. ‘Lectures on ship hydrodynamics—steering and manoeuvrability’, HyA Rept No. Hy-5, 1964.
15.
KroukovskyKorvin B. V. ‘Investigation of ship motions in regular waves’, Trans. Soc. nav. Archit. mar. Engrs195563, 386.
16.
KroukovskyKorvin B. V.JacobsW. ‘Pitching and heaving motions of a ship in regular waves’, Trans. Soc. nav. Archit. mar. Engrs195765, 570.
17.
BishopR. E. D.ParkinsonA. G. ‘On the choice of origin for body axes attached to a rigid vehicle’, J. mech. Engng Sci.196911, 551.
18.
DuncanW. J.The principles of the control and stability of aircraft1952 (Cambridge University Press).
19.
EtkinB.Dynamics of flight1959 (Wiley and Sons, Inc.).
20.
LambH.Hydrodynamics1932 (Cambridge University Press).
21.
NewmanJ. N. ‘Some hydrodynamic aspects of ship manoeuvrability’, Sixth Symp. Naval Hydrodynamics, Washington, 1966, 203.
22.
CumminsW. E. ‘The impulse response function and ship motions’, Schiffstechnik, 19627, 101.
23.
BrardR. ‘A vortex theory for the manoeuvring ship with respect to the history of her motion’, Fifth Symp. Naval Hydrodynamics, Bergen, 1964, 815.
24.
BishopR. E. D.ParkinsonA. G.On the planar motion mechanism used in ship model testing, Phil. Trans. R. Soc. A1970266, 35.
25.
DuncanW. J. ‘Introductory survey’, AGARD Manual19591, 1.
26.
BishopR. E. D.BurcherR. K.PriceW. G. ‘A new representation of fluid forces as applied to ship dynamics’ (to be published).
27.
FroudeW. ‘The papers of William Froude’, Trans. Instn nav. Archit. 1955.
28.
KriloffA. ‘A new theory of the pitching motion of ships on waves and the stresses produced by this motion’, Trans. Instn nav. Archit.189637, 326.
29.
KaplanP. ‘Development of mathematical models for describing ship structural response in waves’, Dept of Navy, NAVSEC Nobs-94322, 1969.
30.
KaplanP.SargentT. P.RaffA. I. ‘An investigation of the utility of computer simulation to predict ship structural response in waves’, Dept of Navy, NAVSEC N00024–67-C-5254, 1969.
31.
PaullingJ. R.WoodL. W. ‘The dynamic problem of two ships operating on parallel courses in close proximity’, Univ. Calif. Pubis Engng, Series No. 189–1, 1962.
32.
Van LeeuwenG. ‘The lateral damping and added mass of an oscillating ship model’, Shipbuilding Laboratory, Delft Technological University, Rept No. 23, 1964.
33.
MotoraS.FujinoM. ‘On the measurement of the stability derivatives by means of forced yawing technique’, J. Soc. nav. Archit. Japan1965118, 48.
34.
VugtsJ. H. ‘The coupled roll-sway-yaw performance in oblique waves’, 12th ITTC, Rome, 1969.
35.
EdaH.CraneC. L. ‘Steering characteristics of ships in calm water and waves’, Trans. Soc. nav. Archit. mar. Engrs196573, 1.
36.
HaddaraM. R. ‘On the stability of ship motion in regular oblique waves’, Int. Ship Progr.197118, 416.
37.
LewisF. M. ‘The inertia of the water surrounding a vibrating ship’, Trans. Soc. nav. Archit. mar. Engrs192937, 1.
38.
TasaiF. ‘Hydrodynamic force and moment produced by swaying and rolling oscillation of cylinders on the free surface’, Research Institute of Applied Mechanics, Kyushu University, vol. IX, 1961, 91.
39.
PorterW. ‘Pressure distributions, added mass and damping coefficients for cylinders oscillating in a free surface’, Univ. Calif. Pubis Engng, Series No. 82–16, 1960.
40.
FrankW. ‘Oscillations of cylinders in or below the free surface of deep fluids’, NSRDC, Washington D. C., Rept No. 2375, 1967.
41.
FrankW.SalvesenN. ‘The Frank close-fit ship motion computer program’, NSRDC, Washington D. C., Rept No. 3289, 1970.
42.
FaltinsenO.A comparison of Frank close-fit method and some other methods used to find two-dimensional hydro-dynamic forces and moments for bodies which oscillate harmonically in an ideal fluid, Det Norske Veritas, Oslo, Rept No. 69–43–S, 1969.
43.
SalvesenN.TuckE. O.FaltinsenO. ‘Ship motions and sea loads’, Trans. Soc. nav. Archit. mar. Engrs197078, 1.
44.
StokerJ. J.Nonlinear vibrations in mechanical and electrical systems1950 (Interscience).
45.
GrimO. ‘Rolischwingungen Stabilität und Sicherkeit im Seegang’, Schiffstechnik19521, 10.
46.
KerwinJ. E. ‘Notes on rolling in longitudinal waves’, Int. Ship Progr.19552, 597.
47.
RosenbergR. M. ‘On the stability of a nonlinear, non-autonomous system’, Second U. S. Congr. Applied Mechanics, Ann Arbor, 1954.
48.
CesariL.Asymptotic behaviour and stability problems in ordinary differential equations1959 (Springer-Verlag, Berlin).
49.
GambillR. A. ‘Stability criteria for linear differential systems with periodic coefficients’, Riv. Mat. Univ. Parma19545, 169.
50.
GambillR. A. ‘Criteria for parametric instability for linear differential systems with periodic coefficients’, Riv. Mat. Univ. Parma19556, 37.
51.
BaileyH. R.CesariL. ‘Boundedness of solutions of linear differential systems with periodic coefficients’, Rat Archive Mech. Analysis1957–581, 246.
52.
GrimO. ‘The ship in a following sea’, DTMB Transl. 3131965.
53.
Du CaneP.GoodrichG. J. ‘The following sea, broaching and surging’, Trans. R. Instn nav. Archit.1962104, 109.
