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.

METHOD TO CALCULATE COMPONENTS OF ADDED MASS

OF SURFACE CRAFTS

PHƯƠNG PHÁP TÍNH TOÁN CÁC THÀNH PHẦN KHỐI LƯỢNG NƯỚC

KÈM CỦA TÀU MẶT NƯỚC

Đỗ Thành Sen, Trần Cảnh Vinh

Trường Đại học Giao thông 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 experi-ment 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 pa-per aims to generalize and introduce a combination method to determine all components of hydrody-namic coefficient of added mass and inertia moment of marine crafts moving in 6 degrees of freedom

Keywords: Hydrodynamic coefficient, added mass and moment of inertia, hydrodynamics

Tóm tắt: Để thiết lập các hệ phương trình vi phân biểu diễn chuyển động của tàu mặt nước trên

hệ thống mô phỏng buồng lái, các tham số của hệ phương trình bao gồm các hệ số động học cần phải được xác định Đối với một tàu cụ thể, các thành phần này có thể thu được từ công tác thực nghiệm Tuy nhiên, đối với các tàu ở giai đoạn thiết kế, cần phải tính toán dựa trên nền tảng lý thuyết Kết quả của các nghiên cứu trước đây cho thấy không có phương pháp hiện hữu duy nhất nào có thể xác định được đầy đủ các hệ số thủy động Bài viết này nhằm khái quát và giới thiệu một phương pháp tổng hợp xác định tất cả các hệ số thủy động của khối lượng và mô men quán tính nước kèm của tàu biển chuyển động trên 6 bậc tự do

Từ khóa: Hệ số thủy động, khối lượng và mô men quán tính nước kèm, thủy động lực học

1 Introduction

When a surface craft moves on water,

the fluid moving around creates forces

effect-ing 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

iner-tia are only generated when a craft

acceler-ates or deceleracceler-ates They are directly

propor-tional to the body’s acceleration and derived

by equation on 6 degrees of freedom (6DOF)

[6]:

F =

⌈

⌈

⌈

⌈

⌈

X

Y

Z

K

M

N⌉

⌉

⌉

⌉

⌉

= MA ẍ = MA×

[

⌈

⌈

⌈

⌈

u̇

ν ẇ

p q̇

ṙ ]

⌉

⌉

⌉

⌉

(1)

MA=

[

⌈

⌈

⌈

⌈

m11 m12 m13

m21 m22 m23

m31 m32 m33

m14 m15 m16

m24 m25 m26

m34 m35 m36

m41 m42 m43

m51 m52 m53

m61 m62 m63

m44 m45 m46

m54 m55 m56

m64 m65 m66]⌉

⌉

⌉

⌉ (2)

Where ẍ=[𝑢 ̇, 𝑣̇, 𝑤̇, 𝑝̇, 𝑞̇, 𝑟̇ ]𝑇 is acceleration matrix; 𝐹 = [𝑋, 𝑌, 𝑍, 𝐾, 𝑀, 𝑁]𝑇is matrix of hydrodynamic forces and moments in 6DOF:

DOF Motion / rotation Velocities Force

moment

1 surge - motion in x direction u X

2 sway - motion in y direction v Y

3 heave - motion in z direction w Z

4 roll – rotation about the x axis p K

5 pitch - rotation about the y axis q M

6 yaw - rotation about the z axis r N

Fig 1 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

non-dimensional coefficient 𝑚̅𝑖𝑗 which is called

hydrodynamic coefficient of added mass

In order to establish differential

equa-tions 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

2 Fundamental theory

Basing on theory of kinetic energy of

fluid mij is determined from the formula:

mij= −ρ ∮ φS i∂φj

Where S is the wetted ship area,  is

wa-ter density, φi is potentials of the flow when

the ship is moving in ith direction with unit

speed Potentials φi satisfy the Laplace

equa-tion [3] The matrix MA totally has 36

com-ponents as derived in formula (2) However,

with marine craft, the body is symmetric on

port - starboard (xy plane), it can be

conclud-ed 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, 6 Moreover, due to symmetry of the

matrix MA, mij = mji Thus, 36 components

of added mass are reduced to 18:

MA=

[

⌈

⌈

⌈

⌈

m 11 0 m 13

0 m 22 0

m 31 0 m 33

0 m 15 0

m 24 0 m 26

0 m 35 0

0 m 42 0

m 51 0 m 53

0 m 62 0

m 44 0 m 46

0 m 55 0

m 64 0 m 66 ]

⌉

⌉

⌉

⌉

(4)

In the past, there were many studies

de-termining the added mass including

experi-ment 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

3 Methods suggested for determining

hydrodynamic coefficients

3.1 Equivalent elongated Ellipsoid

To calculate mij, the craft can be

relative-ly assumed as an equivalent 3D body such as sphere, spheroid, ellipsoid, rectangular, cyl-inder etc For marine surface craft, the most equivalent representative of the hull is elon-gated ellipsoid with c/b = 1 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= mk22 (6)

m33= mk33 (7) ; m44= k44Ixx (8)

m 55 = k 55 I yy(9) ; m 66 = k 66 I zz (10)

Fig 2 Craft considered as an equivalent Ellipsoid

Where:

k11= A0

2−A0 (11) ; k22= B0

2−B0 (12)

k33 = C0

2−C0 (13) ; k44= 0 (14)

k55 = (L2−4T2)

2

(A0−C0) 2(4T 4 −L 4 )+(C0−A0)(4T 2 +L 2 ) 2 (15)

k 66 = (L2−B2)

2

(B0−A0) 2(L 4 −B 4 )+(A0−B0)(L 2 +B 2 ) 2 (16) And:

A 0 =2(1−ee32)[12ln (1+e1−e) − e] (17)

B0= C0 = 1

e 2 −1−e2 2e 3 ln (1+e

1−e) (18) With e = √1 −b2

a 2 = √1 −d2

d and L are maximum diameter and length overall Inertia moment of the dis-placed water is approximately the moment of inertia of the equivalent ellipsoid:

Ixx= 1

120 πρLBT(4T 2 + B 2) (20)

Iyy= 1

120 πρLBT(4T2+ L2) (21)

Izz= 1

120 πρLBT(B 2 + L 2) (22)

𝑥

z

y

𝑎 = 𝐿/2

0

𝑏 = 𝐵/2

𝑐 = 𝑇

The limitation of this method is that the

calculating result is only an approximation

The more equivalent to the elongated

ellip-soid it is, the more accurate the result is

ob-tained Moreover, this method cannot

deter-mine 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

re-sembling the segment of the representative

ship and its added mass can be easily

calcu-lated The added masses of the whole ship

are obtained by integration of the 2D value

over the length of the hull

Fig 3 Craft is divided into sections

Components 𝑚𝑖𝑗 are determined:

m 22 = ∫ mL2 22 (x)dx

L1 (23) m 33 = ∫ m(x)dxL2

m 24 = ∫ mL2 24 (x)dx

L1 (25) m 44 = ∫ mL2 44 (x)dx

L1 (26)

m26= ∫ mL2 22 (x)xdx

L1 (27) m46= ∫ mL2 24 (x)xdx

𝑚35= − ∫ 𝑚𝐿2 33 (𝑥)𝑥𝑑𝑥

𝑚66= ∫ 𝑚𝐿2 22 (𝑥)𝑥 2 𝑑𝑥

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

Lew-is transformation Lew-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 +pσ+σq3) (31)

And the unit semicircle is derived:

σ = e iφ = cosθ + isinθ (32)

Where i = √−1; a 0= T(x)

1+p+q By substi-tuting into the formula (31), descriptive pa-rameters of a cross section can be obtained: {

y = [(1 + p)sinθ − qsin3θ] B(x)

2(1+p+q)

z = −[(1 − p)cosθ + qcos3θ] B(x)

2(1+p+q)

(33)

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)

H(x) = B(x)

2T(x) =1+p+q

β(x) = A(x)

B(x)T(x)= π

4

1−p2−3q2 (1+q) 2 −p 2 (35)

Fig 4 The transformation from x- and ζ –plane.

Parameter θ corresponds to the polar an-gle of given point prior to conformal trans-formation from a semicircle π/2 ≥ θ ≥ - π/2

q =

3

4 π+√(π

4 )2−π

2 α(1−γ 2 ) π+α(1−γ 2 ) − 1 ; p = (q + 1)q (36)

α = β −π

4 ; γ =H−1

H+1 (37)

The components mij(x) of each section are determined by formulas:

m22(x) =ρπT(x)2

2 (1−p)2+3q2 (1−p+q) 2 =ρπT(x)2

2 k22(x) (38)

m33(x) =ρπB(x)2

8 (1+p)2+3q2) (1+p+q)2 =ρπB(x)2

8 k33(x) (39)

m 24 (x) =ρT(x)3

2 1 (1−p+q) 2 {−8

3 P(1 − p) +16

35 q 2 (20 − 7p) + q [4

3 (1 − p) 2 −4

5 (1 + p)(7 + 5p)]}

=𝜌𝑇(𝑥)3

𝑚 44 (𝑥) = 𝜌𝜋𝐵(𝑥)

4

256

16[𝑝 2 (1 + 𝑞) 2 + 2𝑞 2 ] (1 − 𝑝 + 𝑞) 4

=𝜌𝜋𝐵(𝑥)4

256 𝑘44(𝑥) (41) Then, total mij is calculated:

m22= μ1(λ = L

2T )ρπ

2 ∫ T(x)L2 2k22(x)dx

m33= μ1(λ =L

B )ρπ

8 ∫ B(x)L2 2k33(x)dx

m24= μ1(λ = L

2T )ρ

2 ∫ T(x)L2 3k24(x)dx

72

Journal of Transportation Science and Technology, Vol 20, Aug 2016

m44 = μ1(λ =2TL)256ρπ ∫ B(x)L2 4k44(x)dx

m26 = μ2(λ =2TL)ρπ2 ∫ T(x)L2 2k22(x)xdx

m35 = −μ2(λ =L

B )ρπ

8 ∫ B(x)L2 2k33(x)xdx

m46 = μ2(λ =2TL)ρπ2 ∫ T(x)L2 3k24(x)xdx

m66 = μ2(λ = L

2T )ρπ

2 ∫ T(x)L2 2k22(x)x 2 dx

Where μ1(λ), μ2(λ) are corrections related

to fluid motion along x-axis:

μ1(λ) = λ

√1+λ 2 (1 − 0.425 λ

1+λ 2 ) (50)

μ 2 (λ) = k66(λ, q)q (1 + 1

It is noted that specific forms of ships

consisting of re - entrant forms and

asymmet-ric forms are not acceptable for applying

Lewis forms [1], [5]

3.3 Determining remaining

compo-nents

The Equivalent Ellipsoid and Strip

theo-ry method do 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≈ 0

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= m11ZB (52)

m24= m42 = −m22ZB (53)

Thus, the formula to calculate m15 and

m51 is obtained:

m 15 = m 51 = −m 11

m42

m22 (54) 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

hydrody-namic coefficients are represented in the

form of non-dimension:

m ̅ 11 = m11

0.5ρL 2 (55) m ̅ 22 = m22

0.5ρL 2 (56)

m ̅33 = m33

0.5ρL 2 (57) m ̅24= m24

0.5ρL 2 (58)

m ̅15 = m15

0.5ρL 2 (59) m ̅26= m26

0.5ρL 3 (60)

m ̅46 = m46

0.5ρL 3 (61) m ̅55= m55

0.5ρL 4 (62)

m ̅66 = m66

0.5ρL 4 (63) m ̅35= m35

0.5ρL 2 B (64)

m ̅44 = m44

0.5ρL 2 B 2 (65)

3.5 Calculating hydrodynamic coeffi-cients on computer

For numbering the hull frames and cal-culating 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 1 Numbering hull sections

1 10.000 2.927 60.000 0.000 0.000

2 9.750 2.927 57.073 0.240 1.035

3 9.500 2.927 54.146 0.520 0.788

4 9.250 2.927 51.220 0.773 0.736

5 9.000 5.854 48.293 0.905 0.775

6 8.500 5.854 42.439 1.170 0.782

7 8.000 11.707 36.585 1.190 0.886

8 7.000 11.707 24.878 1.190 0.930

9 6.000 11.707 13.171 1.190 0.945

10 5.000 11.707 1.463 1.190 0.960

11 4.000 11.707 -10.244 1.190 0.960

12 3.000 11.707 -21.951 1.190 0.960

13 2.000 5.854 -33.659 1.190 0.960

14 1.500 5.854 -39.512 1.190 0.930

15 1.000 2.927 -45.366 1.170 0.865

16 0.750 2.927 -48.293 1.150 0.790

17 0.500 2.927 -51.220 1.070 0.733

18 0.250 2.927 -54.146 0.933 0.666

19 0.000 1.463 -57.073 0.773 0.505

20 -0.125 1.463 -58.537 0.586 0.503

21 -0.250 0.000 -60.000 0.320 0.927

Numbering values of mapping are calcu-lated and displayed in curves on computer in Fig 5, 6, 7 and 8 The results indicate that the transformation is relatively proper

Fig 5 Curves of B(x) and A(x)

Fig 6 Curves of H(x) and β(x)

Fig 7 Lewis frames of the fore and aft sections.

Fig 8 Results of Lewis transformation mapping of

the sample craft

The calculating results of two methods

are summed up and presented in table 2 The

column “suggested” are the values suggested

for application by combination of two

meth-ods

Table 2 Calculating value - 𝑚 ̅𝑖𝑗.

Basing on the above results, it is

con-cluded that Strip theory method can

deter-mine most component m̅ij with high accuracy

due to equivalent transformation This

meth-od cannot determine component m̅11, m̅55 but

can be solved by considering the ship as an

elongated ellipsoid

As the component m̅15= m̅51, this value

is not so high, the calculation in the formula

(54) is satisfied and acceptable

4 Conclusion

The above-mentioned method can de-termine all 18 remaining components of hy-drodynamic coefficients of added masswhich are necessary to establish the set of differen-tial equations describing the motion of ma-rine 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 map-ping, craft with re-entrant forms is inapplica-ble 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

Insti-tute - Springer, St Petersburg, Russia, pp 51-55, pp 86-88, pp 93-96

[2] EDWARD M LEWANDOWSKI (2004), The

Dynam-ics 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

trans-action, Vol 102, pp 419-445

[5] J.M.J JOURNÉE & L.J.M ADEGEEST (2003),

The-oretical 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

Uni-versity of Science and Technology Trondheim, Nor-way, John Wiley & Sons

[7] TRAN CONG NGHI (2009), Ship theory – Hull

re-sistance and Thrusters (Volume II), Ho Chi Minh city

University of Transport, pp 208-222

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