Method to calculate components of added mass of surface crafts

With a particular ship, these hydrodynamic components can be obtained by experiment. However, at the design stage the calculation based on theory is necessary. Unluckily, previous studies proved that no unique existing methods can determine all hydrodynamic coefficients. This paper aims to generalize and introduce a combination method to determine all components of hydrodynamic coefficient of added mass and inertia moment of marine crafts moving in 6 degrees of freedom.

TẠP CHÍ KHOA HỌC CƠNG NGHỆ GIAO THƠNG VẬN TẢI, SỐ 20 - 08/2016 69 METHOD TO CALCULATE COMPONENTS OF ADDED MASS OF SURFACE CRAFTS PHƯƠNG PHÁP TÍNH TỐN CÁC THÀNH PHẦN KHỐI LƯỢNG NƯỚC KÈM CỦA TÀU MẶT NƯỚC Đỗ Thành Sen, Trần Cảnh Vinh Trường Đại học Giao thơng Vận tải Thành phố Hồ Chí Minh Abstract: For establishing the differential equations to describe the motion of a surface marine craft on bridge simulator system, parameters of equations including hydrodynamic coefficients need to be determined With a particular ship, these hydrodynamic components can be obtained by experiment However, at the design stage the calculation based on theory is necessary Unluckily, previous studies proved that no unique existing methods can determine all hydrodynamic coefficients This paper aims to generalize and introduce a combination method to determine all components of hydrodynamic coefficient of added mass and inertia moment of marine crafts moving in degrees of freedom Keywords: Hydrodynamic coefficient, added mass and moment of inertia, hydrodynamics Tóm tắt: Để thiết lập hệ phương trình vi phân biểu diễn chuyển động tàu mặt nước hệ thống mô buồng lái, tham số hệ phương trình bao gồm hệ số động học cần phải xác định Đối với tàu cụ thể, thành phần thu từ cơng tác thực nghiệm Tuy nhiên, tàu giai đoạn thiết kế, cần phải tính tốn dựa tảng lý thuyết Kết nghiên cứu trước cho thấy khơng có phương pháp hữu xác định đầy đủ hệ số thủy động Bài viết nhằm khái quát giới thiệu phương pháp tổng hợp xác định tất hệ số thủy động khối lượng mơ men qn tính nước kèm tàu biển chuyển động bậc tự Từ khóa: Hệ số thủy động, khối lượng mơ men qn tính nước kèm, thủy động lực học Introduction When a surface craft moves on water, the fluid moving around creates forces effecting to the hull These forces are defined as hydrodynamic forces consisting of inertia forces of added mass, damping forces and restoring tensors Added mass and added moment of inertia are only generated when a craft accelerates or decelerates They are directly proportional to the body’s acceleration and derived by equation on degrees of freedom (6DOF) [6]: u̇ X ν̇ Y ⌈ẇ⌉ ⌈ ⌉ F = ⌈ Z ⌉ = MA ẍ = MA × ⌈ ṗ ⌉ ⌈ ⌉ ⌈K⌉ ⌈ q̇ ⌉ ⌈M⌉ ⌈N⌉ [ ṙ ] m11 m21 ⌈m MA = ⌈m31 ⌈ 41 ⌈m51 [m61 m12 m22 m32 m42 m52 m62 m13 m23 m33 m43 m53 m63 m14 m24 m34 m44 m54 m64 m15 m25 m35 m45 m55 m65 Where ẍ = [𝑢̇ , 𝑣̇ , 𝑤̇ , 𝑝̇ , 𝑞̇ , 𝑟̇ ]𝑇 is acceleration matrix; 𝐹 = [𝑋, 𝑌, 𝑍, 𝐾, 𝑀, 𝑁]𝑇 is matrix of hydrodynamic forces and moments in 6DOF: DOF Motion / rotation Velocities surge - motion in x direction sway - motion in y direction heave - motion in z direction roll – rotation about the x axis pitch - rotation about the y axis yaw - rotation about the z axis u v w p q r Force moment X Y Z K M N (1) m16 m26 m36 ⌉ m46 ⌉⌉ m56 ⌉ m66 ] (2) Fig Motions of craft in 6DOF Where mij is component of added mass in the ith direction caused by acceleration in direction j Each component 𝑚𝑖𝑗 is represent- 70 Journal of Transportation Science and Technology, Vol 20, Aug 2016 ed by a coefficient kij or by a nondimensional coefficient 𝑚 ̅𝑖𝑗 which is called hydrodynamic coefficient of added mass In order to establish differential equations of the craft motion, it is necessary to determine the matrix MA For a particular ship, it can be obtained by experimental methods However, for displaying the craft motion on a simulator system especially in design stage, the hydrodynamic components of the matrix MA have to be calculated by theoretical methods Fundamental theory Basing on theory of kinetic energy of fluid mij is determined from the formula: mij = −ρ ∮S φi ∂φj ∂n dS m13 m15 m22 m24 m26 ⌉ m33 m35 ⌉ m42 m44 m46 ⌉ m53 m55 ⌉ m62 m64 m66 ] m55 = k 55 Iyy (9) ; m66 = k 66 Izz (10) (3) Where S is the wetted ship area,  is water density, φi is potentials of the flow when the ship is moving in ith direction with unit speed Potentials φi satisfy the Laplace equation [3] The matrix MA totally has 36 components as derived in formula (2) However, with marine craft, the body is symmetric on port - starboard (xy plane), it can be concluded that vertical motions due to heave and pitch induce no transversal force The same consideration is applied for the longitudinal motions caused by acceleration in direction j = 2, 4, Moreover, due to symmetry of the matrix MA, mij = mji Thus, 36 components of added mass are reduced to 18: m11 ⌈ m MA = ⌈ 31 ⌈ ⌈ m51 [ 3.1 Equivalent elongated Ellipsoid To calculate mij, the craft can be relatively assumed as an equivalent 3D body such as sphere, spheroid, ellipsoid, rectangular, cylinder etc For marine surface craft, the most equivalent representative of the hull is elongated ellipsoid with c/b = and r = a/b Where a, b are semi axis of the ellipsoid Basing on theory of hydrostatics, m11, m22, m33, m44, m55, m66 can be described [1], [7]: m11 = mk11 (5) ; m22 = mk 22 (6) m33 = mk 33 (7) ; m44 = k 44 Ixx (8) 𝑥 𝑎 = 𝐿/2 𝑏 = 𝐵/2 𝑐=𝑇 y z Fig Craft considered as an equivalent Ellipsoid Where: k11 = k 33 = A0 2−A0 C0 2−C0 B0 2−B0 (11) ; k 22 = (13) ; k 44 = In the past, there were many studies determining the added mass including experiment and theoretical prediction However, no single method can determine all components of the matrix [5] By studying defferent methods introdued by prvious studies, the group of authors combine and suggest a combination method to determine the hydrodynamic coefficients of 18 remaining components Methods suggested for determining hydrodynamic coefficients (14) (L2 −4T2 ) (A0 −C0 ) 2 −A0 )(4T +L ) k 55 = 2(4T4 −L4 )+(C (15) (L2 −B2 ) (B0 −A0 ) 2 −B0 )(L +B ) k 66 = 2(L4 −B4 )+(A (4) (12) (16) And: A0 = B0 = 2(1−e2 ) 1+e [2 ln (1−e) − e] e3 1−e2 1+e C0 = e2 − 2e3 ln (1−e) With e = √1 − b2 a2 = √1 − d2 L2 (17) (18) (19) d and L are maximum diameter and length overall Inertia moment of the displaced water is approximately the moment of inertia of the equivalent ellipsoid: Ixx = 120 πρLBT(4T + B ) Iyy = 120 πρLBT(4T + L2 ) Izz = 120 πρLBT(B + L2 ) (20) (21) (22) TẠP CHÍ KHOA HỌC CƠNG NGHỆ GIAO THƠNG VẬN TẢI, SỐ 20 - 08/2016 The limitation of this method is that the calculating result is only an approximation The more equivalent to the elongated ellipsoid it is, the more accurate the result is obtained Moreover, this method cannot determine component m24; m26, m35; m44, m15 and m51 3.2 Strip theory method with Lewis transformation mapping Basing on this method a ship can be made up of a finite number of transversal 2D sections Each section has a form closely resembling the segment of the representative ship and its added mass can be easily calculated The added masses of the whole ship are obtained by integration of the 2D value over the length of the hull 71 T(x) Where i = √−1; a0 = 1+p+q By substituting into the formula (31), descriptive parameters of a cross section can be obtained: y = [(1 + p)sinθ − qsin3θ] B(x) 2(1+p+q) B(x) { z = −[(1 − p)cosθ + qcos3θ] (33) 2(1+p+q) Where B(x), T(x) are the breadth and draft of the cross section s Parameter p, q are described by means of the ratio H(x) and β(x) B(x) 1+p+q (34) H(x) = 2T(x) = 1−p+q A(x) β(x) = B(x)T(x) = π 1−p2 −3q2 (35) (1+q)2 −p2 Fig The transformation from x- and ζ –plane Parameter θ corresponds to the polar angle of given point prior to conformal transformation from a semicircle π/2 ≥ θ ≥ - π/2 Fig Craft is divided into sections Components 𝑚𝑖𝑗 are determined: L m22 = ∫L m22 (x)dx m24 = L2 ∫L m24 (x)dx L m26 = ∫L m22 (x)xdx (23) L m33 = ∫L m(x)dx L2 ∫L m44 (x)dx (25) m44 = (27) m46 = ∫L m24 (x)xdx (28) 𝐿 𝐿2 ∫𝐿 (26) L 𝑚35 = − ∫𝐿 𝑚33 (𝑥)𝑥𝑑𝑥 𝑚66 = (24) 𝑚22 (𝑥)𝑥 𝑑𝑥 (29) (30) Where mij(x) is added mass of 2D cross section at location xs In practice the form of each frame is various and complex For numbering and calculating in computer Lewis transformation is the most proper solution With this method a cross section of hull is mapped conformably to the unit semicircle (ζ-plane) which is derived [1], [2], [5], [7]: ζ = y + iz = ia0 (σ + p q + σ3 ) σ (31) And the unit semicircle is derived: σ = eiφ = cosθ + isinθ π π π+√( ) − α(1−γ2 ) 4 π+α(1−γ2 ) q= π H−1 α = β − ; γ = H+1 (37) The components mij (x) of each section are determined by formulas: ρπT(x)2 ρπT(x)2 (1−p)2 +3q2 m22 (x) = = k 22 (x) (38) (1−p+q)2 m33 (x) = m24 (x) = ρπB(x)2 (1+p)2 +3q2 ) (1+p+q)2 ρT(x)3 (1−p+q)2 = ρπB(x)2 8 {− P(1 − p) + (39) k 33 (x) 16 q (20 35 − 7p) + q [ (1 − p)2 − (1 + p)(7 + 5p)]} = 𝜌𝑇(𝑥)3 𝑘24 (𝑥) 𝑚44 (𝑥) = 𝜌 (40) 𝜋𝐵(𝑥) 16[𝑝 (1 + 𝑞) + 2𝑞 256 (1 − 𝑝 + 𝑞)4 = 𝜌𝜋𝐵(𝑥)4 𝑘44 (𝑥) 256 2] (41) Then, total mij is calculated: ρπ L2 ∫ T(x)2 k 22 (x)dx L1 L ρπ L m33 = μ1 (λ = B) ∫L B(x)2 k 33 (x)dx L ρ L2 m24 = μ1 (λ = ) ∫L T(x)3 k 24 (x)dx 2T L (32) − ; p = (q + 1)q (36) m22 = μ1 (λ = 2T) (42) (43) (44) 72 Journal of Transportation Science and Technology, Vol 20, Aug 2016 m44 = μ1 (λ = m26 = μ2 (λ = m35 = −μ2 (λ L ρπ L2 ) ∫ B(x)4 k 44 (x)dx 2T 256 L1 L ρπ L2 ) ∫ T(x)2 k 22 (x)xdx 2T L1 L ρπ L = B) ∫L B(x)2 k 33 (x)xdx ρπ L2 ∫ T(x)3 k 24 (x)xdx L1 L ρπ L2 ) ∫ T(x)2 k 22 (x)x dx 2T L1 L 46 m ̅ 46 = 0.5ρL (46) 66 m ̅ 66 = 0.5ρL (48) m66 = μ2 (λ = (49) Where μ1 (λ), μ2 (λ) are corrections related to fluid motion along x-axis: λ √1+λ2 λ (1 − 0.425 1+λ2 ) μ2 (λ) = k 66 (λ, q)q (1 + ) λ2 (50) (51) It is noted that specific forms of ships consisting of re - entrant forms and asymmetric forms are not acceptable for applying Lewis forms [1], [5] 3.3 Determining remaining components The Equivalent Ellipsoid and Strip theory method not determine component m15 The nature of marine surface craft is that m13 is relatively small in comparison with total added mass and can be ignored Thus, m13 = m31 ≈ It is approximately considered that the component m15 and m24 are caused by the hydrodynamic force due to m11 and m22 with the force center at the center of buoyancy of the hull ZB [2] Therefore: m15 = m51 = m11 ZB (52) m24 = m42 = −m22 ZB (53) Thus, the formula to calculate m15 and m51 is obtained: m m15 = m51 = −m11 m42 (54) 22 When m24 and m42 can be obtained by the Strip theory method 3.4 Non - dimensional hydrodynamic coefficients To simplify and to make it convenient for deriving added mass and added moment of inertia in complex equations, the hydrodynamic coefficients are represented in the form of non-dimension: m m m ̅ 11 = 11 (55) m ̅ 22 = 22 (56) 0.5ρL 0.5ρL m 33 m ̅ 33 = 0.5ρL m 15 m ̅ 15 = 0.5ρL m (57) 24 m ̅ 24 = 0.5ρL (59) 26 m ̅ 26 = 0.5ρL m m m (61) 55 m ̅ 55 = 0.5ρL (63) m ̅ 35 = 0.5ρL352 B m m ̅ 44 = (47) m46 = μ2 (λ = 2T) μ1 (λ) = m (45) m44 0.5ρL2 B2 (64) (65) 3.5 Calculating hydrodynamic coefficients on computer For numbering the hull frames and calculating the hydrodynamic coefficients on computer, the authors used a craft model with particulars: L = 120m2; B = 14.76m; T = 6.2m; Displacement = 9.178 MT The craft hull is divided longitudinally into 20 stations with ratio H and β: Table Numbering hull sections No 10 11 12 13 14 15 16 17 18 19 20 21 Sta 10.000 9.750 9.500 9.250 9.000 8.500 8.000 7.000 6.000 5.000 4.000 3.000 2.000 1.500 1.000 0.750 0.500 0.250 0.000 -0.125 -0.250 dx 2.927 2.927 2.927 2.927 5.854 5.854 11.707 11.707 11.707 11.707 11.707 11.707 5.854 5.854 2.927 2.927 2.927 2.927 1.463 1.463 0.000 x 60.000 57.073 54.146 51.220 48.293 42.439 36.585 24.878 13.171 1.463 -10.244 -21.951 -33.659 -39.512 -45.366 -48.293 -51.220 -54.146 -57.073 -58.537 -60.000 H 0.000 0.240 0.520 0.773 0.905 1.170 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.190 1.170 1.150 1.070 0.933 0.773 0.586 0.320 β 0.000 1.035 0.788 0.736 0.775 0.782 0.886 0.930 0.945 0.960 0.960 0.960 0.960 0.930 0.865 0.790 0.733 0.666 0.505 0.503 0.927 Numbering values of mapping are calculated and displayed in curves on computer in Fig 5, 6, and The results indicate that the transformation is relatively proper Fig Curves of B(x) and A(x) (58) (60) (62) Fig Curves of H(x) and β(x) TẠP CHÍ KHOA HỌC CƠNG NGHỆ GIAO THƠNG VẬN TẢI, SỐ 20 - 08/2016 Fig Lewis frames of the fore and aft sections Fig Results of Lewis transformation mapping of the sample craft The calculating results of two methods are summed up and presented in table The column “suggested” are the values suggested for application by combination of two methods Table Calculating value - 𝑚 ̅ 𝑖𝑗 𝑚 ̅ 𝑖𝑗 𝑚 ̅11 𝑚 ̅15 𝑚 ̅ 22 𝑚 ̅ 24 𝑚 ̅ 26 𝑚 ̅ 33 𝑚 ̅ 35 𝑚 ̅ 44 𝑚 ̅ 46 𝑚 ̅ 55 𝑚 ̅ 66 Ellipsoid 0.035 1.167 1.167 0.034 0.034 Lewis 1.113 0.814 0.028 1.440 0.002 0.014 0.092 0.065 Suggested 0.035 -0.026 1.113 0.914 0.028 1.440 0.002 0.014 0.092 0.034 0.065 Basing on the above results, it is concluded that Strip theory method can determine most component m ̅ ij with high accuracy due to equivalent transformation This method cannot determine component m ̅ 11 , m ̅ 55 but can be solved by considering the ship as an elongated ellipsoid As the component m ̅ 15 = m ̅ 51 , this value is not so high, the calculation in the formula (54) is satisfied and acceptable 73 Conclusion The above-mentioned method can determine all 18 remaining components of hydrodynamic coefficients of added mass which are necessary to establish the set of differential equations describing the motion of marine surface crafts in six degrees of freedom used for simulator system The suggested method is not applicable for a hull with port-starboard asymmetry Due to the use of Lewis transformation mapping, craft with re-entrant forms is inapplicable In this case, additional consideration should be taken into consideration to make sure the calculating results are satisfied with allowable accuracies  References [1] ALEXANDR I KOROTKIN (2009), Added Masses of Ship Structures, Krylov Shipbuilding Research Institute - Springer, St Petersburg, Russia, pp 51-55, pp 86-88, pp 93-96 [2] EDWARD M LEWANDOWSKI (2004), The Dynamics Of Marine Craft, Manoeuvring and Seakeeping, Vol 22, World Scientific, pp 35-54 [3] HABIL NIKOLAI KORNEV (2013), Lectures on ship manoeuvrability, Rostock University Universität Rostock, Germany [4] J.P HOOFT (1994), “The Prediction of the Ship’s Manoeuvrability in the Design Stage”, SNAME transaction, Vol 102, pp 419-445 [5] J.M.J JOURNÉE & L.J.M ADEGEEST (2003), Theoretical Manual of Strip Theory Program “SEAWAY for Windows”, Delft University of Technology, the Netherlands, pp 53-56 [6] THOR I FOSSEN (2011), Handbook of Marine Craft Hydrodynamics and Motion Control, Norwegian University of Science and Technology Trondheim, Norway, John Wiley & Sons [7] TRAN CONG NGHI (2009), Ship theory – Hull resistance and Thrusters (Volume II), Ho Chi Minh city University of Transport, pp 208-222 Ngày nhận bài: 27/05/2016 Ngày chuyển phản biện: 30/05/2016 Ngày hoàn thành sửa bài: 14/06/2016 Ngày chấp nhận đăng: 21/06/2016 ... above-mentioned method can determine all 18 remaining components of hydrodynamic coefficients of added mass which are necessary to establish the set of differential equations describing the motion of marine... the representative ship and its added mass can be easily calculated The added masses of the whole ship are obtained by integration of the 2D value over the length of the hull 71 T(x) Where i =... in design stage, the hydrodynamic components of the matrix MA have to be calculated by theoretical methods Fundamental theory Basing on theory of kinetic energy of fluid mij is determined from