MIT2_017JF09_p41

# MIT2_017JF09_p41 - 41 FLOATING STRUCTURE HEAVE AND ROLL 41...

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41 FLOATING STRUCTURE HEAVE AND ROLL 165 41 Floating Structure Heave and Roll Consider the heave and roll response of a two-hull structure with the parameters below: mass (metric tons) 3000 body rotary inertia (kg-m 2 ) 3 × 10 8 beam of each hull b (m) 5 open space between hulls B (m) 20 draft T (m) 10 vessel length L (m) 30 each hull effective added mass A 33 (tons) 750 each hull effective damping coeﬃcient B 33 (Ns/m) 4 × 10 5 The hull is assumed to be uniform in cross-section over the entire length. We are going to study the behavior of this structure in beam waves - that is, moving from port to starboard across the two hulls. We will use the Bretschneider spectrum as in Homework 2, for sea states 2-6. 1. About what are the undamped natural frequencies in heave and in roll? Assume in this problem that the vessel’s center of mass is near the waterline. The square root of waterplane stiffness divided by mass and added mass gives about 0.700 rad/s in heave and 0.875 rad/s in roll. 2. About what are the damping ratios in heave and in roll? We have ζ 0 . 095 in heave and 0.12 in roll; not much damping! 3. Up to what wave frequency is the long-wavelength approximation valid for this prob- lem? Make a plot to verify that you have a ratio of at least three or so, up to the highest frequency you will use in the following calculations. The figure below shows that we are OK up to about two radians per second, considering each hull alone. If we take the whole vessel beam, then we could only do the higher sea states, e.g., ω < 0 . 9 . 4. Taking y = 0 at the port side of the port hull, compute for a range of wave frequencies the phase angle in the incident wave heave force ( F 3 I in the notes) seen at the middle of the port hull and then the middle of the starboard hull. Make and annotate a plot. This is shown in the figure; formulas for each term are given below. The angle for the far hull changes very quickly as the wavelengths grow to and then exceed B . 5. Write out the differential equation that governs the heave motions, under wave distur- bances, and including the incident wave, diffraction, and radiation terms.

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41 FLOATING STRUCTURE HEAVE AND ROLL 166 We have from the formulas in the notes mz + ( B 33 p + B 33 s ) z + ( C 33 p + C 33 s ) z = ( F 3 Ip + F 3 Is ) η ( y = 0) + ( F 3 Dp + F 3 Ds ) η ( y = 0) + ( F 3 Rp + F 3 Rs ) z where the p and s subscripts refer to port and starboard hulls, respectively. The coeﬃcients are: kT e F 3 Ip = ρg [sin ωt (cos kb cos 0) cos ωt (sin kb sin 0)] k kT e F 3 Is = ρg [sin ωt (cos k (2 b + B ) cos k ( b + B )) cos ωt (sin k (2 b + B ) k sin k ( b + B ))] F 3 Dp = e kT/ 2 A 33 p ω 2 [cos ωt cos( kb/ 2) + sin
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