24 A. Mondoro et al. Transfer function, Φm 2 0 10 20 30 40 Vertical bending moment spectrum, SVBM Vertical bending moment, Mp /Mp5 Wave spectrum, SPM 0 1 2 0 1 2 3 4 x 10-3 (b) (a) (c) (d) 0 0.5 1 1.5 2 0 2.5 5.0 x 10-3 0 0.5 1 1.5 2 0 1 2 3 4 Frequency, ω Frequency, ω Ship Speed, V(kt) 5 kt 10 kt 15 kt 20 kt 25 kt 30 kt 35 kt Ship Speed, V 5 kt 10 kt 15 kt 20 kt 25 kt 30 kt 35 kt Ship Speed, V Frequency, ω 1.0 1.1 1.2 1.3 1.4 1.5 Fig. 3.1 The variations in the (a) transfer functionˆm 2, (b) wave spectrumSPM, (c) response spectrumSVBM, and (d)moment Mp/Mp5 to changes in the speed of the ship -1 -0.5 0 0.5 1 0 0.05 0.10 0.15 0.20 0 2 4 6 8 10 -0.2 0 0.2 0.4 0.6 0 10 20 30 40 0.06 0.08 0.10 0.12 0.14 Ship Speed, V(knots) Wave Height, Hs (m) Cos(β) Vertical bending moment, Mp Vertical bending moment, Mp Vertical bending moment, Mp (b) (a) (c) A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 C4 Fig. 3.2 Theoretical variations of bending moment with (a) wave height, (b) ship speed, and (c) heading angle (the theoretical values are shown as black circles and proposed functional forms are fit to each and shown as solid lines) the variation of SPM with V lie on top of each other. The response spectrumSVBM is generated for each value of V using Eq. (3.4) and is shown in Fig. 3.1c. SVBM is then used in Eqs. (3.12) and (3.13) to find the most probable vertical bending moment. The variation of Mp and V is depicted in Fig. 3.1d; the vertical bending moment at each ship speed is compared to the vertical bending moment at ship speed of 5 knots, Mp5. The same procedure is applied for Hs and ˇ but figures are omitted for brevity. 3.3.2 Development of Functional Forms A closed-form function to describe theoretical relationship between the operational condition (i.e., Hs, V, and ˇ) and the response (i.e., Mp) is not readily available. As noted in Sect. 3.1, Hs, V, andˇare assumed to be uncoupled in regards toMp; therefore, Mp (Hs, V, ˇ) can be decoupled into Mp (Hs), Mp (V), and Mp (ˇ). The theoretical variations of bending moment with wave height, ship speed, and cosine of the heading angle are shown in Fig. 3.2a–c respectively. Four functional forms were developed for Mp (Hs), Mp (V), and Mp (ˇ) and are listed in Table 3.1. Polynomial functions were used to describe Mp
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