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 Aw of two 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 × 105ton · m2 . Each hull has an apparent linear damping in the vertical direction of b =60kN · 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 wave description, we use the Bretschneider spectrum; it is given by A −B/ω4 S(ω)= e , where ω5 ωm = modal (or peak) frequency, rad/s B =1.25ω4 ; A =4BES; = H2 /16.m ES 1/3In SeaState 5, we take the modal period as 9.7 seconds, and the significant wave height H1/3 as 3.3m. We 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 wave elevations at the two struts. Hint: use the fact that ζ(t, −L/2) = z(t) − φ(t)L/2, and so on. Solution: We have, using the hint, mz¨ = ρgAw[η(−L/2) − (z − Lφ/2) + η(L/2) − (z + Lφ/2)] − b[( ˙z − L ˙z + L ˙φ/2) + ( ˙ φ/2)] = ρgAw[η(−L/2) + η(L/2) − 2z] − 2bz˙Jφ ¨= ρgAw 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 L2 = ρgAw [η(L/2) − η(−L/2) − Lφ] − bφ.˙2 2� � � � � � 28 FLOATING STRUCTURE IN WAVES 93 Because of cancelations in this symmetric situation, we end up with decoupled equa-tions - i.e., the pitch motion does not affect the heave motion, and vice versa. 2. What are the structure’s natural frequencies in heave and in pitch? Do these seem like a good design? Solution: The natural frequency in heave, obtained from the above equation with no wave excitation (η =0)is 2ρgAw/m =1.98rad/s,oraperiod of 3.2sec. The natural frequency in pitch is ρgAwL2/2J =2.48rad/s,oraperiod of 2.5sec. These are quite fast compared to the frequencies that we expect in big seas - a three-second wave in the open ocean is generally quite small in amplitude, less than one meter. So it seems like a good design from the resonance point of view. 3. Give a general expression for the wave elevation at one strut, as a function of the wave elevation at the other strut. To do this, write down the elevations for only one wave, at frequency ω. You will use the dispersion relation to work out the wavelength λ,and hence derive a frequency-dependent phase angle. Solution: We have 2πL η(L/2) = η(−L/2) cos λ Lω2 = η(−L/2) cos , g where the second expression is found by substituting the dispersion relation. 4. Insert this result into your set of differential equations, and come up with an ODE for the heave being driven by waves (referenced to x = −L/2), and another for the pitch being driven by waves. Note your answers will have a ”weird” term similar to cos(ω2), which you can carry directly into the frequency domain (because the Fourier transform is an integration over time!). Solution: mz¨ +2bz˙ +2ρgAwz = ρgAw(1 + cos(Lω2/g))η(−L/2) L2 L2 L Jφ ¨+ bφ˙+ ρgAwφ = − ρgAw(1 − cos(Lω2/g))η(−L/2). 2 2 2 5. What are the two transfer functions z(jω) φ(jω) and ? η(jω, x = −L/2) η(jω, x = −L/2) Solution: z(jω) ρgAw[1 + cos(Lω2/g)] = η(jω, x = −L/2) −mω2 +2bjω +2ρgAw φ(jω) L ρgAw[1 − cos(Lω2/g)] = − 2 . η(jω, x = −L/2) −Jω2 + L2 2 bjω + L2 2 ρgAw� � � � 28 FLOATING STRUCTURE IN WAVES 94 6. Make a labeled plot of the Bretschneider wave spectrum for these SS5 conditions. See the attached graph. 7. What are the significant heights (double amplitudes) of the heave and the pitch mo-tions? Based on plots of the output spectra, about what are the dominant frequencies of these two motions? Solution: The significant heave height is 3.30m and the significant pitch height is 0.121rad. The output spectra plots (attached) are interesting because the dominant frequency in the heave direction is close to the resonance, around 1.9rad/s,or 3.3 seconds period. On the other hand, the pitch direction has a dominant frequency much lower, at about 0.75rad/s, or 8.4 seconds period - this is closer to the excitation frequency. You see that the effect of the different wavelengths gives rise to many peaks in the responses, and this ultimately controls the output spectra shapes. 8. What are the heave and pitch amplitudes expected to be exceeded in ten minutes? one hour? one day? Solution: For heave the amplitudes that will be exceeded are [2.63 3.06 3.70] meters; for pitch, we find [0.094 0.110 0.134] radians. The formulas are (for the z part): � ∞ Miz = ωiSz(ω)dω 0 ¯M0zTz =2π M2z ¯f(Az)=[1/600 1/3600 1/86400] Hertz Az = −2M0z log f¯(Az)T¯z . %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Heave and pitch response of a large floating structure clear all; wm = 2*pi/9.7 ; % modal frequency of waves, rad/s Hsig = 3.3 ; % significant wave height, m wvec = 0.001:0.001:4 ; % vector of frequencies to consider % make up the Bretschneider spectrum for j = 1:length(wvec), w = wvec(j) ; S(j) = 5/16 * wm^4 / w^5 * Hsig^2 * exp(-5 * wm^4 / 4 / w^4) ; end;28 FLOATING STRUCTURE IN WAVES 95 % check that we got the right formula! disp(sprintf(... ’Square Root of Integral of Area of S: %g; Hsig/4: %g’, ...
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