Systematic Approach to Mathematical Modelling of a Submarine Dynamics

at the Periscope Depth

A0114 Sadko Mandzuka Brodarski Institute

Operating at the periscope depth below a seaway, a submarine is in an

unstable condition. As a free surface is approached, the wave forces

increase, trying to pull the submarine to the surface [1]. To counter

these forces, the submarine's ballast may be changed and/or control

surfaces are used. Because of the stochastic nature of waves, the

manual operation of the submarine for longer periods at the periscope

depth exhausts the submarine control operators. Improved periscope

depth performance is therefore imperative [2,3].

As a submarine approaches the free surface, several complexities are

introduced into the mathematical modelling of its dynamics. The

approach used in this paper is first to establish a dynamic model

appropriate for a deeply submerged submarine at moderate speeds. The

forces and moments resulting from the seaway will then be superimposed

on that model to provide a reasonable approximation to the submarine

motion below waves. In the published submarine periscope depth

control research, [4,5], these disturbances were modelled by the

simple affine form. The main disadvantage of this approach is a poor

correspondence with the real situation. The dynamic model is

inadequate, and there is no phase information between Zwave (t), Mwave

(t) and v (t). The seawave spectra are described in the mathematical

form using different kinds of spectra [6]. Unfortunately, the

equations arising from such descriptions are difficult to manipulate

in analytical investigations. Following the linear theory, using

spectral factorisation techniques, the sea spectrum was modelled by

transfer function representation (colour filter)[7]. The model of the

depth-attenuation factor by the first order lag is a reasonable

choice.A new presented high-frequency mathematical model of a

submarine dynamic at the periscope depth is writen in the form of

color filter. The numerical and simulation examples given in this

paper are based on the mathematical model of the dynamic of new

Croatian Navy midget submarine VELEBIT.


[1] Musker, A. J., Loader, P. R., Butcher, M. C.:"Simulation of a Submarine Under Waves ",

International Shipbuilding Progress, 35(1988) 404, p.389-410.

[2] Papoulias, F. A.: "Shallow water characterization ", Report to Naval Surface Warfare Center,

Carderock Div.,1995.

[3] Tolliver, J. V.:"Studies on Submarine Control for Periscope Depth Operations ",

Master's Thesis, Naval Postgraduate School, Monterey, 1996.

[4] Dumlu, D., Istefanopulos, Y.:"Design of an Adaptive Controller for Submersibles via Multimodal Gain Scheduling ",

Ocean Engineering, 22 (1995)6, p. 593 - 614.

[5] Liciega-Castro, E., Van der Molen, G.: "A submarine depth control system design ",

International Journal of Control, 61 (1995)2, p. 279-308.

[6] Lloyd, A. R. J. M.: "SEAKEEPING:Ship Behaviour in Rough Weather ",

John Wiley & Sons, New York, 1989.

[7] Manduka, S.:"Some characteristics of sea spectrum modelling by coloured filter ",

Proceedings of International Symposium: Waves - Physical and Numerical Modelling, p.833-841, Vancouver,1994.