the sum of a large number of regular sinusoidal waves, each being a solution of the linearised hydrodynamic equations describing water waves. Tese advances provided the impetus to assess the behaviour of a rigid ship travelling in an irregular seaway through analysis of wave disturbance and ship response random processes with the introduction of a statistical language (i.e. mean square value, spectrum etc.), incorporating the role of uncertainty through probabilistic measures, into the profession. Te analysis also showed that there remained an essential role for deterministic mathematical models describing the motions of a ship excited by regular sinusoidal waves and the need for experimental model test data to corroborate the theoretical predictions based on a linear mathematical theory. The development of two-dimensional (i.e. strip theory) and

three-dimensional (i.e. treating the ship as a single rigid body) mathematical models allowed description of ship-wave interaction characteristics in terms of receptances or response amplitude operators, which convey information on motion resonances and ship-wave matching. Teir combination with a sea spectrum (Pierson- Moskowitz, JONSWAP, etc) produces a ship response spectrum embodying the statistical data describing an irregular ship motion response.

Three-dimensional hydrodynamics The linear, three-dimensional models overcame the inherent weaknesses of the simpler strip theory models. Namely, the idealisation of the continuous hull form into a series of two-dimensional sections, failure to represent the actual hull shape and omission of fluid-structure interactions between sections. Boundary element methods provided a better representation of the three dimensional hull through a patchwork of panels distributed over the mean wetted hull surface with a suitable source function (i.e. Rankine, etc) satisfying the linearised free surface wave condition placed on each panel. Mathematical models, based on Green theory, initially adopted a zero speed oscillating source function with forward speed treated as a correction to the zero speed solution. Significant computing effort and power were required to derive information describing the six degrees of freedom responses and wave loads experienced by a ship travelling in a seaway. By utilising translating, pulsating sources the mathematical models were able to account for steady state and oscillatory behaviour of the vessel as well as incorporating forward speed. Although the mathematical models were developed in a rigorous theoretical manner, they remain an approximation to reality because of the assumptions of hull rigidity, disturbances are small and only the calm water wetted hull surface area contributes to the calculations of hydrodynamic pressure, responses, wave loadings, etc. Te latter depend on available computer power and developed numerical schemes of study which transfer solution derivation onto studies of the employed numerical methods, numerical analysis (i.e. stability, convergence, accuracy, etc) and inherent model characteristics (i.e. irregular frequency, etc) rather than the physical modelling of the problem.

Hydroelasticity Te mathematical models developed apply to rigid ships, irrespective of size. However as ships evolved through significant changes of construction, available power, specialist cargoes transported and

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their very large increase in size (i.e. tankers, bulk carriers, etc), the assumption of a rigid ship became untenable. For example, a ship hull, at rest in calm water, is acted upon by gravity and buoyancy forces with the result that it deflects, adopting a particular attitude and distortion. Tese deflections depend on the loading of the ship which, when the ship travels in waves become time dependent fluctuations. Te rigid hull assumption was addressed by adopting fundamental

concepts of sound and vibration theory with the development of hydroelasticity theory modelling the hull as a flexible structure experiencing deflections. The fluid surrounding the hull was treated as an external fluid loading exciting the flexible hull and, in the absence of fluid, the ship examined as a dry beamlike, flexible, non-uniform structure. This allowed determination of unique fundamental structural dynamic properties (i.e. principal mode shapes, natural frequencies, etc) by investigating the free vibration of the undamped free-free structure. Te introduction of a suitable beam theory (e.g. Euler, Timoshenko, etc) or finite element model to describe the structure and development of a wet analysis, dependent on the adopted hydrodynamic theory (i.e. strip theory, source distribution, etc), resulted in generalised, coupled equations of motion to describe a flexible ship hull travelling in an irregular seaway. Tis unified mathematical model incorporates both rigid body motions and distortions since a deflection is a summed weighted (i.e. principal modes) combination of both. Te model treats ship strength and ship dynamics on the same fundamental basis providing a framework to assess, at any position in the hull, the behaviour of a vessel to wave induced responses causing stresses, fatigue, etc. Furthermore, in accordance with the theory of applied mechanics, internal loadings of bending moment, shear force, stress etc at all positions in the hull and frequencies are determined only from the distortions of the hull structure. In recent years, attention has focussed on the detailed idealisation

of a hull structure by utilising finite elements. Tis has perhaps limited the position of the naval architect as a highly skilled theoretical innovator since focus of much theoretical innovation has transferred to the embodiment of knowledge into the description of finite element type. Nevertheless, a realistic detailed idealisation of a ship structure by finite elements requires great modelling expertise of the naval architect, significant computing power and a clear understanding of how the ship structure is expected to behave. Te introduction of finite element analyses into naval architecture has provided the practical means of analysing the fluid-structure interaction behaviour of complex mono-and multi- hull structures of all shapes, sizes and materials, (steel, composites, etc) travelling in an irregular seaway based on an unified, linear hydroelastic mathematical model.

Future developments With increasing computer power, the future ultimate goal of naval architecture research is the replacement of linear mathematical models by physically consistent, rigorously developed mathematical models based on Navier-Stokes equations to simulate, in real time, the operational behaviour of a ship travelling in an irregular seaway. Although practical use of Reynolds Average Navier-Stokes (RANS) models are becoming ever more accessible with increasing speed and capability of computers they require a high level of user expertise. Their everyday use as a design optimisation tool still remains

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