 By Prof. Alaa Mansour, Donald Liu, J. Randolph Paulling

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Extra info for The Principles of Naval Architecture Series: Strength of Ships and Ocean Structures

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The frequency response function H(ω), or the RAO, is a function that gives the vessel response to a regular wave of unit amplitude. For example, if the response under consideration is the bending moment amplitude, then STRENGTH OF SHIPS AND OCEAN STRUCTURES the bending moment can be calculated for the vessel in regular waves of different frequencies and for different vessel speeds and headings. Notice that in this case the ordinate of |H(ω)|2 , it is the square of the bending moment per unit wave amplitude, ζ .

Notice that in this case the ordinate of |H(ω)|2 , it is the square of the bending moment per unit wave amplitude, ζ . In general, the RAOs can be obtained either from calculations using the equations of motion of the ship or by towing tank experiment. Each of these will be discussed brieﬂy in the following paragraphs. The general dynamic equations of motion of a vessel in regular waves can be obtained by applying Newton’s law of motion for a rigid body. If the origin is taken at the center of gravity of the body, then F= d (m · V ) dt (52) M= d (I · ω) dt (53) where V = velocity vector F = force vector m= mass M = moment acting on the body ω = angular velocity vector I = moment of Inertia about the coordinate axes The ﬁrst of these equations gives the three force equations in the x, y, and z directions (surge, sway, and heave equations).

If the mean of the input process is zero, the mean of the output process is also zero. 4. If the input is an ergodic process, the output is also an ergodic process. Notice that if the input process is narrow band, the output is not necessarily a narrow band process. Over a short period, ocean waves can be assumed a stationary normal random process with zero mean. The process can be completely characterized by the spectral density, Sx (ω). The area under the spectrum is related to the mean square of the process, therefore certain averages such as average wave height, average of 1/3 highest waves, and so on, can be determined.