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| In the field of [[shipbuilding|ship design]] and design of other floating structures, a '''response amplitude operator''' ('''RAO''') is an engineering statistic, or set of such statistics, that are used to determine the likely behavior of a ship when operating at sea. Known by the acronym of ''RAO'', response amplitude operators are usually obtained from models of proposed ship designs tested in a [[model basin]], or from running specialized [[computational fluid dynamics|CFD]] computer programs, often both. RAOs are usually calculated for all [[ship motions]] and for all [[Direction (geometry)|wave headings]].
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| == Usage ==
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| RAOs are effectively [[transfer function]]s used to determine the effect that a [[sea state]] will have upon the motion of a ship through the water, and therefore, for example, whether or not (in the case of cargo vessels) the addition of cargo to the vessel will require measures to be taken to improve [[Instantaneous stability|stability]] and prevent the cargo from shifting within the vessel. Generation of extensive RAOs at the design phase allows shipbuilders to determine the modifications to a design that may be required for safety reasons (i.e., to make the design robust and resistant to capsizing or sinking in highly adverse sea conditions) or to improve performance (e.g., improve top speed, fuel consumption, stability in rough seas). RAOs are computed in tandem with the generation of a [[hydrodynamic]] [[database]], which is a model of the effects of water pressure upon the ship's hull under a wide variety of flow conditions. Together, the RAOs and hydrodynamic database provide (inasmuch as this is possible within modelling and engineering constraints) certain assurances about the behavior of a proposed ship design. They also allow the designer to dimension the ship or structure so it will hold up to the most extreme sea states it will likely be subjected to (based on [[sea state]] statistics).
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| == RAOs in ship design ==
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| Different modelling and design criteria will affect the nature of the 'ideal' RAO curves (as plotted graphically) being sought for a particular ship: for example, an ocean cruise liner will have a considerable emphasis placed upon minimizing accelerations to ensure the comfort of the passengers, while the stability concerns for a naval warship will be concentrated upon making the ship an effective weapons platform.
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| # Finding the forces on the ship when it is restrained from motion and subjected to regular waves. The forces acting on the body are:
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| ## The [[Froude–Krylov force]], which is the [[pressure]] in the undisturbed waves integrated over the wetted surface of the ship.
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| ## The [[Diffraction]] forces, which are pressures that occur due to the disturbances in the water because of the ship being present.
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| # Finding the forces on the ship when it is forced to [[oscillate]] in still water conditions. The forces are divided into:
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| ## [[Added mass]] forces due to having to accelerate the water along with the ship.
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| ## [[Damping]] forces due to the oscillations creating outgoing waves which carry energy away from the ship.
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| ## [[Restoring force]]s due to bringing the buoyancy/weight equilibrium out of balance.
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| In the above "Ship" must be interpreted widely to also include other forms of floating structures. The obvious problem in the above method is the neglection of viscous forces which contribute heavily in modes of motion like [[Ship motions|surge]] and [[Ship motions|roll]].
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| On a computer the above algorithm was first introduced by using [[strip theory]]. Today strip theory is still used if the need for fast calculations outweigh the need for precise results and the ship designer knows the limitations of strip theory. More advanced programs that are used today (such as WAMIT, SESAM WADAM, MOSES and ANSYS AQWA ) may not use the exact algorithm outlined above and some may also include the effects of viscosity. The insight into the forces that governs the seakeeping behavior of a ship gained from the above are of course still valid.
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| == Calculating the RAO ==
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| The RAO transfer function is only defined when the ship motions can be assumed to be [[Linear differential equation|linear]]. The above forces can then be assembled into an [[Harmonic oscillator|equation of motion]]:
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| :<math>\big[ M + A(\omega) \big] \ddot x + B(\omega) \dot x + C x = F(\omega)</math>
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| Where <math>x</math> is a degree of freedom (e.g. a vector of rigid body motions), <math>\omega</math> is the oscillation frequency, <math>M</math> is the structural mass and inertia <math>A(\omega)</math> is the added mass (frequency dependent), <math>B(\omega)</math> is the linear damping (frequency dependent), <math>C</math> is the restoring force coefficient (similar to stiffness matrix in a structural dynamic function) and <math>F(\omega)</math> is the harmonic excitation force proportional to <math>x</math> and the wave height <math>\zeta_a</math>.
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| This can be solved for <math>x</math> and the RAO is then:
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| :<math>\mathrm{RAO}(\omega) = \frac{x}{\zeta_a} = \frac{F_0}{C - (M+A(\omega)) \omega^2 + i B(\omega) \omega}</math>
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| where <math>F_0</math> is the linear excitation force complex amplitude per wave height. The RAO is a frequency dependent and complex function (the <math>i</math> in the above expression is the [[imaginary unit]]). It is common to only consider the absolute value of the RAO if the phase between the excitation and the ship motions are irrelevant.
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| It is common to add a linearised viscous damping term, <math>B_v</math> to account for the strong non-linearity of the damping force, especially in the [[ship motions|roll motion]]. This term is for simplicity often taken as a fraction of the [[Damping ratio|critical damping]], <math>B_v = \xi\ B_\text{crit}</math>. The expression is then:
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| :<math>\big[ M + A(\omega) \big] \ddot x + \big[ B(\omega) + B_v \big] \dot x + C x = F(\omega)</math>
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| where:
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| :<math>B_v = \xi\ B_\text{crit} = \xi\ 2 \sqrt{(A(\omega) + M) C}</math>
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| A good simplification is to use the infinite frequency added mass, <math>A_\infty = \lim_{\omega \to \infty} A(\omega)</math> in the above expression to find a frequency independent critical damping value.
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| == See also ==
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| * [[Ship motions]]
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| * [[Ship design]]
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| * [[Naval architecture]]
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| ==References==
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| * [http://www.ultramarine.com/hdesk/runs/samples/sea_keep/doc.htm Ultramarine Inc. web page] illustrating RAO curves and describing their uses (note: contains content aimed at ship design professionals)
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| * {{cite book
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| | last = Faltinsen
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| | first = O. M.
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| | title = Sea Loads on Ships and Offshore Structures
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| | publisher = [[Cambridge University Press]]
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| | year = 1990
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| | isbn = 0-521-45870-6 }}
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| [[Category:Shipbuilding]]
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| [[Category:Naval architecture]]
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