This paper describes the development of strip theory to predict quickly with sufficient accuracy ship motions in wave taking into account nonlinear effects caused by interactions ship-wave. Recently, many ship design tools are available, however a quick tool is still needed to use in preliminary design stage. The developed Strip Theory was applied to Ferry motions in heading wave. Here, the Ferry is assumed moving with velocity 6.2m/sec. in wave height 1m and 4m based on actual condition. The Ferry motions were computed in range wave length 0.5Lwl to 2.0mLwl. The research results in wave height 1m and 4m show the heaving amplitudes increase from 0.5Lwl to 0.6Lwl and then decrease to 0.7Lwl, furthermore they increase from 0.9Lwl to 2.0Lwl in increasing weave length, respectively. Moreover, pitching amplitude tends to decrease from 0.5Lwl to 0.6Lwl and then it increase until 2.0Lwl. The nondimensional heaving motions are greater than 1.0 from 1.6Lwl to 2.0Lwl. This means that the wave load affects heaving amplitude bigger than wave amplitude and this condition becomes a concern point whereas the pitching response is small caused by wave impact in the all wave length. In addition, the present results were compared with CFD results as well. Our present results indicate good agreement with CFD results. In our future works, the present results will be compared with experimental results as well

Baso, Suandar

Hasanuddin University Repository

 
D-  9 
 
 
New Strip Theory Approach to Ship Motions Prediction 
 
 
Suandar BASO
1
, Syamsul ASRI
1
, Rosmani
1
, Lukman BOCHARY
1
, L.A.H PRATAMA
2 
1)
 Naval Architect Department, Engineering Faculty, Hasanuddin University 
2) 
Naval Architect Department, Engineering Faculty, Hasanuddin University 
Jl. Perintis Kemerdekaan Km. 10 Tamalanrea, 90245. 
 E-mail :andar_baso@yahoo.co.id 
 
Abstract 
 
This paper describes the development of strip theory to predict quickly with sufficient accuracy ship motions in wave taking 
into account nonlinear effects caused by interactions ship-wave. Recently, many ship design tools are available, however a 
quick tool is still needed to use in preliminary design stage. The developed Strip Theory was applied to Ferry motions in 
heading wave. Here, the Ferry is assumed moving with velocity 6.2m/sec. in wave height 1m and 4m based on actual 
condition. The Ferry motions were computed in range wave length 0.5Lwl to 2.0mLwl. The research results in wave height 
1m and 4m show the heaving amplitudes increase from 0.5Lwl to 0.6Lwl and then decrease to 0.7Lwl, furthermore they 
increase from 0.9Lwl to 2.0Lwl in increasing weave length, respectively. Moreover, pitching amplitude tends to decrease 
from 0.5Lwl to 0.6Lwl and then it increase until 2.0Lwl. The nondimensional heaving motions are greater than 1.0 from 
1.6Lwl to 2.0Lwl. This means that the wave load affects heaving amplitude bigger than wave amplitude and this condition 
becomes a concern point whereas the pitching response is small caused by wave impact in the all wave length. In addition, 
the present results were compared with CFD results as well. Our present results indicate good agreement with CFD results. 
In our future works, the present results will be compared with experimental results as well. 
 
Keywords: Strip Theory, Ship Motions, Heave Amplitude, Pitch Amplitude 
 
 
1. Introduction 
Ship accidents at the sea are still common happen. This accident usually causes financial loss, 
claims human lives, and leads to protracted legal wrangling. Therefore this accident takes more 
attentions many stakeholders. The poor design, navigation, bad weather, fire, and other human 
errors can lead to ship accident [1]. Study interaction ship-wave has been developed recently. As 
known that a strongly interaction between ship-wave can generate nonlinear effects suffered ship 
body [2]. Nonlinear effects are important consideration in ship design. Some phenomena due to 
nonlinear effects are experienced by a ship such as over rolling, slamming, water on deck, 
whipping and wave breaking. These can affect ship performance. Therefore, a ship should be 
designed properly by taking into account nonlinear effects to avoid an accident. In ship design, ship 
response is importantly predicted. Many ship motion predictions use linear and simple assumption. 
However, nonlinear effects in some prediction tools cannot be neglected and are rarely involved 
such as Slender-body theory [3], enhanced unified theory [4], 3D panel method [5] and Rankin 
panel method [6]. In additions, some computational fluid dynamics (CFD) tools are available. In our 
previous research, the ship motions in nonlinear wave were predicted by using CFD, hybrid 
Eulerian scheme with Lagrangian particle [2,7,8,9,10], which the tool can be used to predict ship 
motions taking into account all phenomena with high accuracy. However, almost the all of CFD tool 
needs time cost. Moreover, Ship design requires quick and accurate tool for making proper 
decision. Recently, it is possible to use ship motions programs based on the linear strip theory in 
ship design work. The designer needs quick and easy computational tools for convenient in use 
and accurate to interpret. Based on some reasons, our present research is focused on 
development of strip theory taking into account nonlinear effects caused by interaction between 
ship-wave with sufficient accuracy and quick. 
 
2. Stirp Theory Method 
Strip theory, the forces on and motions of a three-dimensional floating body, can be determined by 
using results from two-dimensional hydromechanics coefficients and exciting wave loads. These 
values will be integrated over the ship length numerically. The ship is considered to be a rigid body. 
Strip theory considers a ship to be made up of a finite number of transverse two dimensionalstrips 
or cross sections, which are rigidly connected to each other. It is assumed that the problem of the 
motions of this floating body in waves is linear. Then, the differential equations will be solved to 
obtain the motions. 
A ship, co-ordinate systems as shown in Figure 1, advances in wave and in the positive x-direction 
with constant speed U. Regular waves with absolute frequency 𝜔0and direction 𝛽are incident on 
the ship. The body frame is fixed to the ship and is described by three translations and rotations 
 
D-  10 
 
 
relative to the hydrodynamic frame. The second right-handed co-ordinate system X(𝑥0 , 𝑦0 , 𝑧0) is 
fixed in space. The ( 𝑥0 , 𝑦0 )-plane lies in the still water surface, 𝑥0 is directed as the wave 
propagation of direction𝛽  and 𝑧0 is directed upwards. Another right-handed co-ordinate system 
O(x,y,z) is directed by x in the direction of the forward ship speed U, y in the lateral port side 
direction and z vertically upwards. The origin O lies vertically above or under the time-averaged 
position of the center of gravity G. The (x,y)-plane lies in the still water surface. Then, the third 
right-handed co-ordinate system G 𝑥𝑘 , 𝑦𝑘 , 𝑧𝑘 is connected to the ship with its origin at G where 𝑥𝑘 in 
the longitudinal forward direction, 𝑦𝑘 in the lateral port side direction and 𝑧𝑘upwards. In still water, 
the (𝑥𝑘 , 𝑦𝑘)–plane is parallel to the still water surface.  
The equations of motions for six degrees of freedom of a ship in wave caused by any external 
loads are based on Newton's second law (F=m.a). Based on co-ordinate system as shown in 
Figure 1, the ship motion equations are given as follows: 
 
 
 
 
 
 
 
 
 
 
 
 
 
Here, a coupled equations of heave and pitch motions, ship motions in heading wave, are firstly 
distinguished. Then, the supposing the wave moves in the negative x0 direction with an angle 
relative to the ship's speed U, the relevant rigid body velocities with a constant forward velocity 
and heaving and pitching in head waves are given by:  
   𝑀𝑘𝑗 + 𝐴𝑘𝑗  𝑥 𝑗 + 𝐵𝑘𝑗 𝑥 𝑗 + 𝐶𝑘𝑗 𝑥𝑗  = 𝐹𝑘 1 
6
𝑗 =1
 
  For 𝑘 = 1,2,3,4,5,6 
where:  
k= 1,3,5; coupled surge, heave and pitch  
k= 2,4,6; coupled sway, roll and yaw  
𝑥 𝑗  Acceleration of harmonic oscilation in direction 𝑗 
𝑥 𝑗  Velocity of harmonic oscilation in direction 𝑗 
𝑥𝑗  Displacement of harmonic oscilation in direction 𝑗 
Fk Harmonic exciting wave force or moment in direction k  
Mkj Solid mass or inertia coefficient  
Akj Hydrodynamic mass or inertia coefficient  
Bkj Hydrodynamic damping coefficient  
Ckj Spring coefficient  
 
2.1 Ship-body boundary condition 
𝑈  = 𝑈𝑖 ; Forward speed        (1) 
𝜉  3 𝑡 = 𝜉 3 𝑡 𝑘   ; Heave velocity       (2) 
𝜉  5 𝑡 = 𝜉 5 𝑡  ; Pitch angular velocity      (3) 
 
where, 𝑛  (𝑡)is the time-dependent unit vector to the instantaneous position of the ship-body 
defined by 
𝑛  𝑡 𝑛 0 + 𝜉 
 
5𝑛 0and𝑛0 is its value when the ship is at rest defined by𝑛0 =  𝑛1, 𝑛2, 𝑛3. 
The body boundary condition is derived for small heave and pitch motions. The nonlinear 
boundary condition on the exact position of the ship hull is stated as follow: 
 
𝜕Φ
𝜕𝑛
= 𝑉  . 𝑛           (4) 
 
D-  11 
 
 
where,      
Φ = Φ3 + Φ5         (5) 
 
𝑉  = 𝑈𝜄 + 𝐾   𝜉 3 − 𝑋𝜉 5         (6) 
where, Φ is unsteady velocity as the linear superposition of heave and pitch components. The 
total rigid-body velocity is the sum of vertical velocity due to heave and pitch.  
The principal assumption of strip theory is certain components of the radiation and diffraction 
potentials. 
  
2.2 Radiation problem 
The ship is forced to oscillate in heave and pitch in advancing at a speed U. The variation of the 
flow in the x-direction is more gradual than its variation around a ship section and then 
expressed by: 
 
𝜕
𝜕𝑋
Φ ≪
𝜕Φ
𝜕𝑌
,
𝜕Φ
𝜕𝑌
         (7) 
 
The 2D equation applies for each strip location at station X. The heave and pitch potentials are 
given as follows:        
 
 
∂2
𝜕𝑋 2
+
𝜕2
𝜕𝑋 2
 𝛷𝑗 = 0, in fluid j = 3,5       (8) 
Then, ship-hull condition at station-X for the heave and pitch potentials are given as follows: 
 
Φ = 𝑅𝑒 𝜙𝑒𝑖𝜔𝑡           (9) 
  
𝜙 = Π3Χ3 + Π5Χ5         (10) 
 
𝜕Χ3
𝜕𝑛
= 𝑖𝜔𝑛3 ;      on S B         (11) 
 
𝜕Χ5
𝜕𝑛
= −𝑖𝜔𝑛3 + 𝑈𝑛3  ;  on S B        (12) 
where, 
 
𝜕2
𝜕𝑌 2
+
𝜕2
𝜕𝑍2
 𝑋𝑗 = 0, in fluid domain       (13) 
 
Then, it is defined by the normalized potential 𝜓3as follows:  
𝜕𝜓 3
𝜕𝑛
= 𝑛3 ;onS B          (14) 
 
𝜕2
𝜕𝑌 2
+
𝜕2
𝜕𝑍2
 𝜓𝑗 = 0, in fluid domain       (15) 
In order to obtained the solution of the 2 two dimension (2D) along a ship which consits of 
several sections used to describe hull form can be carried out 2D panel method. Moreover, the 
heave and pitch potentials as given by the eqautions as follows:   
𝜙 = Π3Χ3 + Π5Χ5  
 
𝑋3 = 𝑖𝜔𝜓3         (16) 
 
𝑋5 =  −𝑖𝜔𝑋 + 𝑈 𝜓3        (17) 
 
Φ = Re 𝜙𝑒𝑖𝜔𝑡  =  Φ3 + Φ5       (18) 
The 2D heave added-mass and damping coefficients due to a section oscillation are expressed 
by: 
𝑎33 𝜔 −
𝑖
𝜔
𝑏33 𝜔 = 𝜌  𝜓3𝑛3𝑐(𝑥) d𝑙      (19) 
where the encounter frequency is given by:  
𝜔 =  𝜔0 − 𝑈
𝜔0
2
𝑔
𝑐𝑜𝑠𝛽         (20) 
Based on integration along a ship and over each cross section at station –X the 3D added mass 
and damping coefficients for heave and pitch are expressed by as follows: 
𝐴33 =  𝑑𝑋𝑎33 (𝑋)𝐿         (21) 
 
 
D-  12 
 
 
𝐵33 =  𝑑𝑋𝑏33 (𝑋)𝐿         (22) 
 
𝐴35 = −  𝑑𝑋𝐿 𝑎33 −
𝑈
𝜔2
𝐵33         (23) 
 
𝐴53 = −  𝑑𝑋𝐿 𝑎33 +
𝑈
𝜔2
𝐵33         (24) 
 
𝐵35 = −  𝑑𝑋𝐿 𝑏33 +
𝑈
𝜔2
𝐴33        (25) 
 
𝐵53 = −  𝑑𝑋𝐿 𝑏33 −
𝑈
𝜔2
𝐴33        (26) 
 
𝐴55 =  𝑑𝑋𝑋
2
𝐿
𝑎33 +
𝑈2
𝜔2
𝐴33       (27) 
 
𝐵55 =  𝑑𝑋𝑋
2
𝐿
𝑏33 −
𝑈2
𝜔2
𝐵33         (28) 
 
2.3 Diffraction problem  
The total potential is relatively to the ship frame and it can be written as follow: 
Φ = Φ𝐼 + Φ𝐷 = 𝑅𝑒  𝜙𝐼 + 𝜙𝐷 𝑒
𝑖𝜔𝑡         (29) 
where, 
𝜙𝐼 =
𝑖𝑔𝐴
𝜔0
𝑒𝐾𝑍−𝑖𝐾𝑋𝑐𝑜𝑠𝛽 −𝑖𝐾𝑌𝑠𝑖𝑛𝛽 ; 𝐾 =
𝜔0
2
𝑔
      (30) 
 
Then, the diffraction potential is defined by: 
𝜙0 =
𝑖𝑔𝐴
𝜔0
𝑒−𝑖𝐾𝑋𝑐𝑜𝑠𝛽 ; 𝜓7 =  𝑋, 𝑌; 𝑍        (31) 
 
The 3D linear free surface condition for𝜙𝐷 can be written as follows: 
 𝑖𝜔 − 𝑈
𝜕
𝜕𝑋
 
2
𝜙𝐷 + 𝑔
𝜕𝜙𝐷
𝜕𝑍
= 0; 𝑍 = 0      (32) 
The same condition is satisfied by𝜙𝐼 
 𝑖𝜔 − 𝑈
𝜕
𝜕𝑋
 
2
𝜙𝐼 + 𝑔
𝜕𝜙𝐼
𝜕𝑍
= 0; 𝑍 = 0       (33) 
Therefore, the heave and pitch exciting forces follow by simply pressure integration as follows: 
𝑋𝑖 𝑡 = 𝑅𝑒 𝕏𝑖𝑒
𝑖𝜔𝑡  ; 𝑖 = 3,5        (34) 
𝕏3 = 𝜌𝑔𝐴  𝑑𝑋𝑒
−𝑖𝐾𝑋𝑐𝑜𝑠𝛽
𝐿
 (𝜓𝐼 + 𝜓7)𝑛3𝑑𝑙𝐶 𝑋      (35) 
𝕏5 = 𝜌𝑔𝐴  𝑑𝑋(−𝑋)𝑒
−𝑖𝐾𝑋𝑐𝑜𝑠𝛽
𝐿
 (𝜓𝐼 + 𝜓7)𝑛3𝑑𝑙𝐶 𝑋      (36) 
where the approximation𝑛5 = −𝑋𝑛3 
Finally the coupled heave and pitch equation can be expressed by: 
 −𝜔2 𝐴𝑖𝑗 + 𝑀𝑖𝑗  + 𝑖𝜔𝐵𝑖𝑗 + 𝑋𝑖 𝜋𝑗 = 𝕏𝑖 𝜔0 ;  𝑖, 𝑗 = 3,5    (37) 
 
   𝑀𝑘𝑗 + 𝐴𝑘𝑗  𝑥 𝑗 + 𝐵𝑘𝑗 𝑥 𝑗 + 𝐶𝑘𝑗 𝑥𝑗  = 𝐹𝑘
𝑗 =3,5
 
 
3. Result And Discussion 
Here, the new Strip Theory, the developed method, was applied to heave and pitch motions of 
the Ferry in heading waves. The main dimensions and body lines plan are presented in Table 1 
and Figure 2, respectively. For the wave conditions, the ship is simulated into two wave height 
cases and 16 wave length cases. The wave heights 𝐻𝑤 are 1m and 4m based on actual 
conditions where the conditions were experienced by the ship. Then, the wave length  𝛼   is 
defined by the water line length of the ship Lwl because it is consider that the wave length is 
relatively to the ship length. Therefore, the wave lengths are from 0.5Lwl to 2.0Lwl. The ship hull 
is divided into 20 sections.  
Figures 3 and 4 show the examples of time history of heave in wave height 1m and wave length 
0.5Lwl and pitch amplitudes, respectively. The heave amplitude 𝜉3and pitch amplitude 𝜃 are 
defined in center gravity point. From those figures, these show that the heave and pitch 
oscillations are quite stable in time period. Moreover, the amplitudes which are obtained are 
(38) 
 
D-  13 
 
 
acceptable displacement because by referring some research results, small wave length impact 
generates low motion response. The overall amplitude were obtained and then used to analyze 
to interpret the ship behavior using nondimensional parameter.  
In the field of ship design, nondimensional parameter of ship response is used to determine the 
behavior of a ship when moving at sea state. In addition, this parameter can be used to interpret 
safety reason to allow modify ship design. Nondimensional heave motion 𝜉3
∗
and pitch motion𝜉3
∗
 
are defined by motion amplitude and wave height. The equations are given as follows: 
𝜉3
∗ =
𝜉3
𝐻𝑤
          (39) 
𝜉5
∗ =
𝜉5
(𝐻𝑤 𝑘)
         (40) 
where, k is wave number. 
 
 
 
 
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figures 5a,b show the nondimensional heave motions over the wave length in both wave height 
cases. From those figures, the heave motions increase initially when 0.5Lwl and then decrease 
until 0.8Lwl. The heave motions gradually increase when wave length greater than 0.9Lwl in 
increasing wave length. The tendency is similar for the both cases and the response 
displacement increases linear from 1m to 4m wave length. This means that any developed Strip 
Theory produces linear result. The maximum heave amplitude is in 2.0Lwl that are 0.5628m in 
1m wave length and 2.2517m in 4m. 
 
 
 
Fig. 3 Time history of heave amplitude 
 
D-  14 
 
 
 
 
 
 
 
 
 
 
  
 
 
 
 
 
 
 
Fig. 4 Time history of Pitch amplitude 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 5 The nondimensional heaving motions; a).Hw=1 meter, b). Hw=4 meter 
 
In additions, figures 6a,b show nondimensional pitch motions. The tendencies of pitch motions 
are the same with heave motions for both cases. However, the pitch motions decrease small in 
0.5Lwl and then increase gradually when greater than 0.6Lwl in increasing wave length. The 
maximum heave amplitude is given in 2.0Lwl where 0.0842rad. for 1m wave length and 
0.3187rad. for 4m.  
Furthermore, the present results are compared with CFD results.  
 
 
 
 
 
 
 
 
D-  15 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
Fig. 6 The nondimensional pitching motions; a).Hw=1 meter, b). Hw=4 meter 
 
4. Conclusion 
 
The strip theory is developed by considering nonlinear wave effects. It is capable to compute in 
predicting heave and pitch motions of a ship. The developed strip theory was applied 
successfully to the Ferry. The present results indicate acceptable and quite accurate. In order to 
increase the useful of this developed strip theory it would be applied in many kinds of ship 
types. Then, the developed strip will be enhanced to handle other ship motions 6DOF. However, 
this developed method will be modified perfectly more to become robust tool. The comparison 
results between the present results and CFD results show quite a little different. Therefore, we 
have some efforts to perform experiment in our future works to validate the present results. 
 
Acknowledgment  
 
This research was sponsored by the LP2M Hasanuddin University with contract No. 110/UN4-
42/LK.26/SP-UH/2013. The authors are sincerely grateful to Prof. Sudirman as the Head of 
LP2M Hasanuddin University. Also we would like to thank very much to all reviewers for their 
fruitful criticisms and recommendations. Finally, I, as the first author, would like to extend my 
acknowledgments to all authors for their help in completing the research report and this paper. 
 
References 
Iskandar B.H., (2007), “Lokasi ketenggelaman kapal ro-ro dan cakupan data GWS”, Seminar 
Nasional Membedah Kelaikan dan Keselamatan Kapal Ro-ro Pengangkut 
Penumpang, in bahasa Indonesia. Universitas Dharma Persada, Jakarta  
 
D-  16 
 
 
Baso S. and Mutsuda H., (2012), “Fluid Structure Interaction with Nonlinear Free Surface Flow: 
Marine Engineering Applications Using Eulerian Schme with Lagrangian Particles”, 
Text Book LAP LAMBERT Academic Publishing, ISBN 978-3659-29826-4.  
Newman, J.N. (1964). “A slender-body theory for ship oscillation in waves”, Jour. of Fluid 
Mechanics, vol.18, pp.602-618. 
Kashiwagi M., Kawazoe K., dan Inada M., (2000), “A study on ship motion and added resistance 
in waves”, Journal of the Kansao Society of Naval Architects, No.234, pp.85-84. 
Kim T. dan Kim Y., (2011), “Numerical Study on Difference-Frequency-Induced Parametric  
Roll Occurrence”, International Journal of Offshore and Polar Engineering , Vol. 21, No. 2, pp. 
111–120.  
Ommani B. dan Faltinsen O., (2001), “Study on Linear 3D Rankine Panel Method for Prediction 
of Semi-Displacement Vessels Hydrodynamic Characteristics at High Speed”, 11th 
International Conference on Fast Sea Transportation FAST 2011, Honolulu, Hawaii, 
USA  
Baso S and Mutsuda H, (2012), “Nimulation on Hydroelastic Behaviors Of A Ship Caused by 
Impact Using Eulerian Scheme With Langrangian”, Local conference of the SENTA 
2012, Fakultas Teknologi Kelautan, Institut Teknologi Sepuluh november, pp.166-178.  
Mutsuda H. dan Doi Y., (2009), “Numerical Simulation of Dynamic Response of Structere by 
Wave Impact Pressure using an Eulerian Scheme with Lagrangian Particles”, Proc. Of 
the ASME 28 th Internatioanal Conference on Ocean, Offshore and Arctic 
Engineereing, Hawaii, OMAE2009-79736.  
Baso S., Matsuda H., Kurihara T., Kurokawa T., Doi Y. dan Shi J., (2011), “ An Eulerian Schme 
with Lagrangian Particles for Evaluating of Seakeeping performance of Ship in 
Nonlinear Wave”, International Journal of Offshore and Polar Engineereing Vol.21, 
No.2, pp.103-110.  
Baso S., Matsuda H., Kawakami K., Hashihira K., dan Doi Y., (2011), “Numerical Study on 
Nonlinear Hydrolactic and Hydrodynamic Effect on Floating Body Using Eulerian 
Schme With Lagrangian particle”, Proc. Of the 21 st International Offshore and Polar 
Engineereing 2011, Hawaii USA. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 


New Strip Theory Approach to Ship Motions Prediction

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New Strip Theory Approach to Ship Motions Prediction

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