University:
Massachusetts Institute of Technology
Course:
2.29 | Numerical Marine Hydrodynamics (13.024)
Academic year:

2022
Views:

288

Pages:

18
Author:

Cirnfusrz0

Top View F_a = p_s * A_s F_a = -A_s * m_s F_a = A_s * m_s Side View C_1 M_1 I C_2 M_2 I C_3 M_3 cg y_1 = m_1l_1m_s y_2 = m_2l_2m_s M_1 = -A_sl_1m_s M_2 = -A_sl_2m_s (M_1A_s)m_s + (b_1l_1y_1 + c_1l_1)m_s + (b_2l_2y_2 + c_2l_2)m_s = F_ee^(iw*t) (b_1l_1 + b_2l_2)m_s + (c_1l_1 + c_2l_2)m_s = F_ee^(iw*t) C_s ST3C F3(x) = ge e-ikz cos Ngelky sins ek dl N3 is the vertical component of the 2D normal to the section. N₂ is the horizontal component of the 2D normal. ẞ is the wave propagation angle. de is the element of arc length around the section. hs is the sectional diffraction force. h3(x) = woe-ikz.cos Wo= gk 3 (23 + N2 sin 3)eikysin kadl wow + ku cos ẞ 3 is the velocity potential for a 2D cylinder of shape C oscillating in heave. is the solution to V² 3 = 0 subject to the boundary condition for heave motion. It can be obtain by several ways including panel methods. Thus, to do the the longitudinal integrals (de), one must know the 2D hy- drostatic terms and the 2D added mass, damping, and velocity potential for heave. SWAY, ROLL AND YAW EQUATIONS [-w² (Α22 + M) + ίω Β22] (2+1-w² (Α21 - Mze) + B24] (+/-w² A26 + tw B26] 66 = F2 [-w²(A42-M2c) + iwB12) 62 + /-w² (Ass +14) + iwBs + Cas) C + [-²(146 - 146) + B46 6 = Fa [-w² A62 + iwB62) (2+/-w² (A64-146) + iw Bes] 6s+/-w² (A66 +16) + twB6666 = Fa All the coefficients can be determined from the 2D sectional sway and roll added mass and damping, the 2D sectional potentials for sway and roll and the hydrostatic roll restoring force. For all the five motions considered, the response at the resonant frequency is largely controlled by the wave generation damping (B coefficients) except for roll where the damping at resonance is dominated by the viscous damping. Therefore, for strip theory to give accurate results for roll, an estimate for the viscous damping coefficient must be added to B55 ST4A SIMULATIONS OF SHIP MOTIONS IN RANDOM SEAS The complete problem includes effects of waves coming from all directions. Here, for simplicity and clarity we will consider long-crested random waves coming from one direction. The "system functions", ξ;(w), are complex numbers, dependent on frequency. For each one, the magnitude is the ratio of the sinusoidal motion amplitude to the wave amplitude. The phase is the phase lead of the motion with respect to the wave elevation at the origin of the coordinate system. Suppose we have a wave spectrum, Se(w). The wave elevation at the origen of the ship, or offshore structure, at the origin of the coordinate system, can be simulated as: ((t) = Σ 2η (πω) n-N where: Zn 1 δε (πδω) δω Within the restrictions of linear theory, each ship motion can be simulated in the specified random wave field as: N 7(t) = Σξ (ηδω) Ζηεί (now)t ーーN

Week 12 Notes: Oscillating Rigid Objects

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