MIT2_017JF09_p28

# MIT2_017JF09_p28 - 28 FLOATING STRUCTURE IN WAVES 28 92...

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28 FLOATING STRUCTURE IN WAVES 92 28 Floating Structure in Waves We consider the pitch and heave dynamics of a large ﬂoating structure in a random sea. You can consider this a two-dimensional problem. The structure has two main, identical struts that pierce the water: each has area A w of hundred square meters, and their centers are separated by a distance L of ﬁfty meters. The mass center of the structure is at the mid-point. The mass m is 1000 tons, and the mass moment of inertia about the centroid is J =4 . 0 × 10 5 ton · m 2 . Each hull has an apparent linear damping in the vertical direction of b =60 kN · s/m . The horizontal motion of the structure is nearly zero. The vertical excitation force exerted at each of the struts may be approximated as the stiﬀness (provided by the strut’s water-plane area) times η ζ ,where η is the wave elevation at the location of the strut’s center, and ζ is the vertical displacement of the strut. Make linearizations where needed. Note we do not take into account any added mass forces in this problem. Also, we assume that the mass center is low on the water, so that the pitching moment is generated exactly by the net loss of ﬂotation on one side and/or the increase of ﬂotation on the other. For the description, we use the Bretschneider spectrum; it is given by A B/ω 4 S ( ω )= e , where ω 5 ω m = modal (or peak) frequency, rad/s B =1 . 25 ω 4 ; A BE S ; = H 2 / 16 . m E S 1 / 3 In SeaState 5, we take the modal period as 9.7 seconds, and the signiﬁcant height H 1 / 3 as 3.3m. assume that the waves are all traveling in the same direction, from negative x toward positive x . 1. Write a pair of diﬀerential equations, that express the heave motion of the center of mass (say z ( t )), and the pitch motion (say φ ( t )), in terms of the elevations at the struts. Hint: use the fact that ζ ( t, L/ 2) = z ( t ) φ ( t ) L/ 2 , and so on. Solution: have, using the hint, mz ¨ = ρgA w [ η ( L/ 2) ( z Lφ/ 2) + η ( L/ 2) ( z + Lφ/ 2)] b [( ˙ z L ˙ z + L ˙ φ/ 2) + ( ˙ φ/ 2)] = ρgA w [ η ( L/ 2) + η ( L/ 2) 2 z ] 2 bz ˙ ¨ = ρgA w L [ η ( L/ 2) + ( z Lφ/ 2) + η ( L/ 2) ( z + Lφ/ 2)] + 2 L b [( ˙ z L ˙ z + L ˙ φ/ 2) φ/ 2)] 2 L L 2 = ρgA w [ η ( L/ 2) η ( L/ 2) ] bφ.

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MIT2_017JF09_p28 - 28 FLOATING STRUCTURE IN WAVES 28 92...

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