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1 19 横傾斜状態で航行する船の流体力微係数と操縦性 * 正会員安川宏紀正会員平田法隆 * Maneuverability and Hydrodynamic Derivatives of Ships Traveling in Heeled Condition by Hironori Yasukawa, Member Noritaka Hirata, Member Summary Oblique towing test and circular motion test with changing heel angle were carried out to capture the hydrodynamic forces acting on ship models of container ship, pure car carrier and ferry. Based on the measured forces, the hydrodynamic derivatives with respect to lateral velocity (v), yaw rate (r) and heel angle (φ) were obtained for surge force, lateral force, yawing moment and roll moment. Using the derivatives obtained, ship maneuverability traveling in heeled condition was discussed based on a linear theory presented in this paper. As a result, we made clear the maneuvering characteristics of heeled ships. When the ship travels in heeled condition to starboard side, she turns to port side. Such characteristic depends on the order of magnitude and the plus(minus) sign of the hydrodynamic derivatives with respect to heel angle, Y φ and N φ. In case of the ferry, the turning strength per unit heel angle was about 1/3 comparing with that per unit rudder angle. There is a trend that the turning performance improves with increasing the heel angle. This is due to the course instability appears with increasing the heel angle. The course stability is deeply related to nonlinear derivatives with respect to heel angle, such as Y vφφ, Y rφφ, N vφφ and N rφφ Hirano and Takashina 1) 2) 3) * 原稿受理平成年 8 月 817 日 17 4) Kim 5) PMM 6) (Circular Motion Test; CMT) Araki 7) ONRT 3 CMT

2 2 日本船舶海洋工学会論文集第 17 号 213 年 6 月 Fig.1 O X Y Z X Y Z 8) o xyz x y z o X x ψ φ x, y u, v z r U u 2 + v 2 β tan 1 ( v/u) δ Table 1 Principal particulars of ship models L (m) B (m) d (m) Volume (m 3 ) C b x G/L F n (a) X (b) x U v β u (C) o r y Fig. 2 Ship models O ψ δ φ o z y Y (X) (Y ) (N) (K) X, Y x y N,K z x Fig. 1 Coordinate systems β r φ 2. 2 Table 1 () () 2 2 () L B d 3 Volume C b x G 3 Fig.2 3 Table 1 (F n) β 2 deg 2 deg, 5 deg r.2.2,.1 φ 15, 1, 5, deg 4 x ρ( ), L, d, U 2 2) X v r φ 2 X vvvv Y, N, K 1 +3

3 横傾斜状態で航行する船の流体力微係数と操縦性 21 r r 2 X = R + X vvv 2 + X vrv r + X vφv φ +X rφr φ + X φφφ 2 + X vvvvv 4 (1) Y = Y vv + Y r r + Y φφ + Y vvvv 3 + Y vvrv 2 r +Y vvφv 2 φ + Y vφφv φ 2 + Y rφφr φ 2 (2) N = N vv + N rr + N φφ + N vvvv 3 + N vvrv 2 r +N vvφv 2 φ + N vφφv φ 2 + N rφφr φ 2 (3) K = K vv + K rr + K φφ + K vvvv 3 + K vvrv 2 r +K vvφv 2 φ + K vφφv φ 2 + K rφφr φ 2 (4) R Y v, Y r 2. 4 Fig.3 (β) (Y ) (N ) (K ) r (φ) Y φ N φ ( ) 3 (2) (4) 3 Y N K β 15, 2 deg φ 15 deg K (, ) (4) K Fig.4 (r ) Y, N β Y r Y φ φ Y N 3 Fig.5 (β) X r β X β X X β> β < 3 Table 2 X vr Y r Table 2 Hydrodynamic derivatives based on midship position X vv X vr + m y X vvvv X vφ X rφ X φφ Y v Y r m x Y vvv Y vvr Y φ Y vvφ Y vφφ Y rφφ N v N r N vvv N vvr N φ N vvφ N vφφ N rφφ K v K r K vvv K vvr K φ K vvφ K vφφ K rφφ Y φ N φ Table 3 Y φ N φ S175 SR18 LPS -twin 2 -sngl 1 ONRT 2 2 (m ) Fig.6 m v, r, φ Y φ, N φ ( v = r = ) φ Y φ N φ

4 22 日本船舶海洋工学会論文集第 17 号 213 年 6 月 β = 2(deg) β = 2(deg) β = 2(deg) β = 1(deg) β = 1(deg) β = 1(deg) β = (deg) β = (deg) β = (deg) β = 1(deg) β = 1(deg) β = 1(deg) N β = 2(deg) β = 2(deg) β = 1(deg) β = (deg) K.1.8 β = 1(deg) β = 2(deg) β = 2(deg) N β = 2(deg) β = 1(deg) β = (deg) β = 1(deg) β = 2(deg) K N β = 2(deg) β = 2(deg) K.1.8 β = 1(deg) β = (deg) β = 1(deg) β = 2(deg) β = 1(deg) β = 2(deg) β = 2(deg) β = 1(deg) β = (deg) β = 2(deg) β = 1(deg) (deg) β = 1(deg) β = 2(deg) β = 2(deg) β = 1(deg) (deg) β = 1(deg) β = 2(deg) Fig. 3 Lateral force, yaw moment and roll moment coefficients with different drift angles versus heel angle (r = )

5 横傾斜状態で航行する船の流体力微係数と操縦性 r =.2 r =.1 r = r =.1 r =.2.1 r =.2 r =.1 r = r =.1 r = N.2.1 r =.2 r =.1 r = r =.1 r = N N r =.2 r =.1 r = r =.1 r = r =.2 r =.1 r = r =.1 r = r =.2 r =.1 r = r =.1 r = Fig. 4 Lateral force and yaw moment coefficients with different non-dimensional yaw rate versus heel angle (β = ) X.5 φ = (deg) φ = 5 (deg) φ = 1 (deg) φ = 15 (deg).3.4 X.5 φ = (deg) φ = 5 (deg) φ = 1 (deg) φ = 15 (deg) X φ = (deg) φ = 5 (deg) φ = 1 (deg) φ = 15 (deg) Fig. 5 Surge force coefficients with different heel angle versus hull drift angle (r = )

6 24 日本船舶海洋工学会論文集第 17 号 213 年 6 月 Y φ N φ.81, % Y φ, N φ Y φ N φ N φ Table 3 Comparison of Y φ, N φ and z v/d name m Y φ N φ z v/d H- 1) S175 2) F-LPS 3) F- 3) twin 5) sngl 5) ONRT 7) Y φ.2.1 ONRT S175 H F F LPS m N φ ONRT S175 H F F LPS m K Y β K Y K Y z v = K v/y v (5) z v (z v) (L) K Y z v Table 3 z v/d (m ) Fig.8 F- z v/d Fig.8 -sngl z v/d.5.7 z v Y K Fig.7 K.5.4 φ = (deg) φ = 5 (deg).3 φ = 1 (deg) φ = 15 (deg).2 mean Fig. 6 Comparison of Y φ and N φ versus mass coefficient m Fig. 7 Comparison of K versus Y for various heel angles K Y Fig.7 Y K (φ) Table 2 Table 4 Y vφφ, Y rφφ, N vφφ, N rφφ (φ) Y v, Y r, N v, N r Y vφφ,

7 横傾斜状態で航行する船の流体力微係数と操縦性 25 z v / d S175 sngl twin F LPS m Fig. 8 Comparison of z v/d versus mass coefficient m 3. ( ) 3. 1 ( ) m + m x u ( ) m + m y v r = X (6) ( ) m + m y v + ( ) m + m x u r m yα φ z = Y (7) 2, N vφφ, 2 Y rφφ 3 N rφφ 3 Y vvφ, N vvφ Y N v v Y vvφ N vvφ v ( ) 3 Y vvφ N vvφ Table 4 Comparison of nonlinear derivatives related to heel angle Y vφφ N vφφ Y rφφ N rφφ Y vvφ N vvφ ( ) I z + J z ṙ = N G (8) ( ) I x + J x φ m yα z v = K (9) sway-roll u v φ m I x,i z x z m x,m y x y J x,j z x z α z X, Y, N G,K (1) (4) (δ) (v, r ) U (φ ) (7)(8) ( ) m + m y v + ( ) m + m x r = Y φφ + ( ) Y v + Y vφφφ 2 v + ( ) Y r + Y rφφφ 2 r +Y δ cos φ δ (1) ( ) I z + J z ṙ = N φφ + ( N v + N vφφφ) 2 v + ( N r + N rφφφ 2 ) r + N δ cos φ δ (11)

8 26 日本船舶海洋工学会論文集第 17 号 213 年 6 月 Y δ, N δ φ (Y φφ N φφ ) v r Y vφφ, Y rφφ, N vφφ, N rφφ φ (1)(11) v T 1T 2 r + ( T 1 + T 2) ṙ + r = T 3 δ + S δδ + S φφ (12) T 1T 2 =(m + m y)(i z + J z)/c (13) ] T 1 + T 2 = [(m + m y)n r +(I z + J z)y v /C (14) T 3 =(m + m y)n δ cos φ /C (15) S δ =(Y δ N v Y v N δ) cos φ /C (16) S φ =(Y φn v Y v N φ)/c (17) C = Y v N r (Y r m m x)n v (18) Y v = Y v + Y vφφφ 2 Y r = Y r + Y rφφφ 2 N v = N v + N vφφφ 2 N r = N r + N rφφφ 2 (19) S φ S δ 9) S φ δ ( T 3 = ) t = r = r = (12) [ r =(S φφ + S δδ ) 1 T 1 T 1 T e t /T T ] 2 T 1 T 2 e t /T 2 (2) (2) ( ) T 1,T 2 (13)(14) C 3. 2 Table 1 3 (,, ) 3 Table 5 φ ( ) φ Table 5 Hydrodynamic derivatives on maneuvering for analysis and course stability criterion Y v Y r m x Y δ Y φ Y vφφ Y rφφ N v N r N δ N φ N vφφ N rφφ C(φ = ) Table 5 C.25L m y =.126, J z = φ 15 deg 2 deg N r Fig.9 Y v, Y r m x, N v, φ (19) Yv, Nr Y r m x, N v Y vφφ, Y rφφ, N vφφ, N rφφ Y v, N r Fig.1 C φ C 1 Yv Nr, C 2 (Yr m m x)n v C C 1 C 2 C 1 C Fig.11,, C

9 横傾斜状態で航行する船の流体力微係数と操縦性 27 linear derivatives Fig N v * Y v * Y r * m x 1N r * φ (deg) Change of linear derivatives versus heel angle C C(=C 1 C 2 ) C 2 C φ (deg) Fig. 1 Change of C and the components C 1 and C 2 versus heel angle ΔC φ = ΔC = 3 ΔC ΔC φ (deg) (17) (Y φn v Y v N φ) Y φ, (Y φn v Y v N φ) S δ C φ = S φ S δ S φ 1 S δ 3 3 deg 1 deg Fig.13 T 1,T 2 T 1,T 2 T 1 S φ & S δ S δ S φ φ (deg) Fig. 12 T 1 & T Change of S φ versus heel angle T 1 1 T φ (deg) Fig. 11 Change of ΔC versus heel angle for 3 ships Fig. 13 Change of T 1 and T 2 versus heel angle Fig.12 S φ S δ S φ 4. 3

10 28 日本船舶海洋工学会論文集第 17 号 213 年 6 月 3 ( ) Y φ, N φ 1/3 Y vφφ, Y rφφ, N vφφ, N rφφ 1) Hirano, M. and Takashina, J. : A Calculation of Ship Turning Motion Taking Coupling Effect due to Heel into Consideration, 59 (198), pp ) 15 (1982), pp ) 2 (26), pp ) 93 (1997), pp ) Kim, Y.-G., Kim, S.-Y., Kim, H.-T., Lee, S.-W. and Yu, B.-S.: Prediction of the Maneuverability of a Large Container Ship with Twin Propellers and Twin Rudders, J. Marine Science and Technology, Vol.12, No.3 (27), pp ), ( ), 19 (25), pp ) Araki, M., Sada-Hosseini, H., Sanada, Y., Tamamoto, K., Umeda, N. and Stern, F.: Estimating Maneuvering Coefficients using System Identification Methods with Experimental, System-Based, and CFD Free-Running Trial Data, Ocean Engineering, 51(212), pp ) 173 (1993) pp ) 424 (1964), pp ) 13 (211), pp ),,, (211) pp ) 146 (1979), pp A1. Table 1 3 (,, ) (CMT) ( )

11 横傾斜状態で航行する船の流体力微係数と操縦性 Y* Y* r =.2 r =.1 r = r =.1 r =.2 N* N*.5 r =.2 r =.1 r = r =.1 r =.2 Fig.A1 CMT CMT 11) Table A1 Table 2 Y v N r 12) 2 2 Fig. A1.2 r =.2 r =.1 r = r =.1 r = r =.2 r =.1 r = r =.1 r =.2 CMT results: lateral force and yawing moment coefficients for and (2)(3) φ = Y = Y vv + Y r r + Y vvvv 3 + Y vvrv 2 r + Y δ δ (A1) Table A1 Hydrodynamic derivatives based on midship position including the effect of propeller and rudder Y v Y r m x Y vvv Y vvr Y δ N v N r N vvv N vvr N δ N = N vv + N rr + N vvvv 3 + N vvrv 2 r + N δδ (A2)

Fig. 3 Coordinate system and notation Fig. 1 The hydrodynamic force and wave measured system Fig. 2 Apparatus of model testing

Fig. 3 Coordinate system and notation Fig. 1 The hydrodynamic force and wave measured system Fig. 2 Apparatus of model testing The Hydrodynamic Force Acting on the Ship in a Following Sea (1 St Report) Summary by Yutaka Terao, Member Broaching phenomena are most likely to occur in a following sea to relative small and fast craft