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Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2013 Article ID 853793 15 pageshttpdxdoiorg1011552013853793
Research ArticleInitiating a Mathematical Model for Prediction of6-DOF Motion of Planing Crafts in Regular Waves
Parviz Ghadimi Abbas Dashtimanesh and Yaser Faghfoor Maghrebi
Department of Marine Technology Amirkabir University of Technology Hafez Avenue No 424 PO Box 15875-4413 Tehran Iran
Correspondence should be addressed to Parviz Ghadimi pghadimiautacir
Received 19 March 2013 Revised 16 June 2013 Accepted 10 July 2013
Academic Editor Viktor Popov
Copyright copy 2013 Parviz Ghadimi et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Nowadays most of the dynamic research on planing ships has been directed towards analyzing the ships motions in either 3-DOF (degrees of freedom) mode in the longitudinal vertical plane or in 3-DOF or 4-DOF mode in the lateral vertical plane Forthis reason the current authors have started a research program of describing the dynamic behavior of planing ships in a 6-DOFmathematical model This program includes the developing of a 6-DOF computer simulation program in the time domain Thistype of simulation can be used for predicting the response of these planing vessels to the environmental disturbances during high-speed sailing In this paper the development of themathematical model will be presented Furthermore a discussion will be offeredabout the use of these static contributions in a time domain simulation for modeling the behavior of planing crafts in regular waves
1 Introduction
Prediction of planing craft motion is one of the maincomputational challenges in marine engineering Due to theinvolved computational time computational fluid dynamicsis very expensive Experimental works are also very costlyTherefore several researchers have tried to present mathe-matical models which are very easy and economical In thelast decades two branches ofmathematical models have beendeveloped which are in two or three degrees of freedom Firstbranch was developed by Savitsky [1] Savitskyrsquos model hasnot been developed extensively For example it cannot beapplied to the planingmotion in irregular waves and it is alsodifficult to use it to obtain time domain simulation Howeversecond branch which was developed by Martin [2] has beenimplemented by many authors In reality Martinrsquos [2] modelwas in frequency domain Zarnick [3] worked on this modeland performed time domain calculations Later Zarnick [4]developed Martinrsquos model even further for planing craftmotion in irregular waves He compared his results withthe experimental findings of Fridsma [5 6] and found hismodel to be in favorable agreement with the experimentaldata However Zarnickrsquos model had some restrictions whichhad been resolved by some researchers In fact after several
decades a model which was initially developed by Zarnick isstill a reliable tool for planing craft motion
Keuning [7] extended Zarnickrsquos [3 4] model to incorpo-rate a formulation for the sinkage and trim of the ship at highspeeds He also studied the hydrodynamic lift distributionalong the length of the ship with nonlinear added massand wave exciting force in both regular and irregular wavesHicks et al [8] expanded the full nonlinear force andmoment equations of Zarnick [3 4] in a multivariable Taylorseries They replaced equations of motion by a set of highlycoupled constant-coefficient ordinary differential equationsvalid through third order Akers [9] summarized the semi-empirical method three-dimensional panel method andtheir advantages and drawbacks dealing with planing hullmotion analysis He reviewed in detail the two-dimensionallow aspect ratio strip theory developed by Zarnick [3 4]Akers [9] modeled the added mass coefficients based on anempirical formula that is a function of deadrise angle Garmeand Rosen [10] presented a similar time domain analysis ofsimulating a planing hull in head seas which is differentfrom the classical Zarnickrsquos model in precalculation schemeof hydrostatic and hydrodynamic coefficients
Later Grame and Rosen [11] improved his model byadding a reduction function based on model tests and
2 International Journal of Engineering Mathematics
published model data for the near-transom pressure Thisreduced the pressure near the stem gradually to zero at thestem vanDayzen [12] extended the originalmodel developedby Zarnick [3 4] and later extended by Keuning [7] to threedegrees of freedom of surge heave and pitch motion inboth regular and irregular head seas The simulations can becarried out with either a constant forward speed or constantthrust He also validated the results by experimental dataof two models and found his model very sensitive to thehull geometry More recently Sebastianii et al [13] developedprevious studies to combine the roll heave and pitch degreesof freedom
In the current paper a mathematical model based on themodels of Zarnick [3] and Sebastianii et al [13] is developedIn all previous studies 2119889 + 119905 theory based symmetric wedgewater entry has been usedHowever the present paper tries todevelop a mathematical model based on asymmetric wedgewater entry which leads to various forces and momentsActually this mathematical model can be considered as afirst step in extending Zarnickrsquos model to six degrees offreedom and the authors are well aware of the fact that thismathematicalmodelmust be graduallymodified step by step
2 Equations of Motions
Planing motions can be divided into two main parts linearand angular motion Moreover various forces and momentsincluding hydrodynamics force hydrostatic force weightand wave effects must be considered in equations of motionHowever no aerodynamic forces are investigated Based onNewtonrsquos second law governing equations of motions can bewritten as
119898 = 119865119909(119905) = 119883
119903(119905) + 119883
119908(119905) + 119883
119891(119905)
119898 119910 = 119865119910(119905) = 119884
119903(119905) + 119884
119908(119905) + 119884
119891(119905)
119898 = 119865119911(119905) = 119885
119903(119905) + 119885
119908(119905) + 119885
119891(119905)
119868119866119909119909
= 119872119909(119905) = 119870
119903(119905) + 119870
119908(119905) + 119870
119891(119905)
119868119866119910119910
120579 = 119872119910(119905) = 119875
119903(119905) + 119875
119908(119905) + 119875
119891(119905)
119868119866119911119911
= 119872119911(119905) = 119877
119903(119905) + 119877
119908(119905) + 119877
119891(119905)
(1)
where subscripts 119903 119908 and 119891 denote hydrostatic hydrody-namic and wave force and moment 119865 and 119872 also indicateforce andmomentwhile119898 119868 and 119905 aremassmoment inertiaand time respectively Equations (1) are written based onthe shown coordinate system To solve equations of motionsforce and moment must be calculated To calculate force andmoment various theories such as 2119889 + 119905 momentum andadded mass theory become necessary
3 2119889 + 119905 Theory
In a ship-fixed coordinate system 25D theory means thatthe two-dimensional equations are solved together withthree-dimensional free surface conditions If the attention is
focused on an Earth-fixed cross-plane one will see a time-dependent problem in 2D cross-plane when the vessel ispassing through it Accordingly the theory is also called 2119889+
119905 theory In an Earth-fixed coordinate system a prismaticplaning vessel of trim angle 120591 (up to 20 degrees) is movingthrough an Earth-fixed cross-plane with speed 119880
119904 as shown
in Figure 1 At time 119905 = 1199050 the cross-section is just above the
free surface at time 119905 = 1199051the cross-section is penetrating
the free surface and at time 119905 = 1199052 flow separates from
the chine line Therefore one can see a process where aV-shaped section enters the water surface in this cross-plane by a speed of 119881 = 119880
119904120591 However it must be noted
that the developed mathematical model has some limitationespecially at high speed Trim angle cannot exceed 20 degreesand the wavelength must also be larger than the hull length
Same procedure can be defined for penetration of anasymmetric wedge into the free surface This means that sideforce can also exist This side force leads to roll momentand yaw moment Therefore various motions of planing hullcan be taken into account Based on this definition twodifferent coordinate systems should be considered (as shownin Figures 2 and 3) Generally three fundamental aspects dueto inclination of section which are shown in Figure 4 can beconsidered [13]
(i) Nonsymmetrical action of fluid on boat the boat inoblique sea undergoes different actions of fluid in itsport and starboard side due to different absolutewavevelocities and relative boat motions relevant to roll
(ii) Nonsymmetry of the section impacting against waterdue to roll motion the section which impacts againstwater is not symmetrical starboard and port sides areconsidered separately with their ldquoequivalent deadriseanglerdquo which is the resultant of the local geometricaldeadrise and the roll angle in order to compute theadded mass terms
(iii) Nonsymmetrical submerged volume geometry sub-merged volume and wet surface are no longer sym-metrical affecting hydrostatics and in general forceapplication points
Finally it is concluded that the force acting on the hull mustbe calculated section by section for starboard side and portside separately
4 Regular Wave Theory
In the present computationalmodel wave forces are obtainedby neglecting diffraction forces (only Froude-Krylov forcesare considered) It is also assumed that the wave excitationis caused by the instantaneous wetted surface and by thevertical component of the wave orbital velocity at the surface119908119911 The influence of the horizontal component of wave
orbital velocity on both the horizontal and vertical motions isneglected because this velocity is considered to be relativelysmall in comparison with the forward speed of the craft Thenormal velocity119881 and the velocity component parallel to thekeel119880 can be written as functions of the craftrsquos forward speed
International Journal of Engineering Mathematics 3
-
Us
t = t0
t = t2
t = t1
V
120591
Earth-fixed cross-plane
Figure 1 Demonstration of 2119889 + 119905 concept [14]
G
BL
WLo
zz998400
Us
x998400
120579
Figure 2 Coordinate system 1 [13]
WL
G
Z
x
Y
120595
120577
o
Figure 3 Coordinate system 2 [13]
heave pitch and vertical component of wave orbital velocity[7] as in
119880 (120577 120585) = 119862119866
(119905) cos 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)] sin 120579 (119905)
119881 (120585 119905) = 119862119866
(119905) sin 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)]
times cos 120579 (119905) minus 120579 (119905) 120585
(2)
FSX
FDX
120573y
z
120573 + 120595
Figure 4 Force acting on an inclined wedge [13]
For regular waves the wave elevation of a linear deep waterwave [13] is
120585 (119909 119910 119905) = 119886 cos (119896119909 cos 120583 + 119896119910 sin 120583 minus 120596119890119905 + 120576) (3)
where 119886 is the wave amplitude 119896 is the wave number and 119888 isthe wave celerity
5 Force Acting on the Hull
As mentioned earlier the numerical model employed herefor prediction of planing motion utilizes a modified 2119889 +
119905 theory with momentum theory The vessel is consideredto be composed of a series of 2D wedges and the threedimensional problem is subsequently solved as a summationof the individual 2D slices The forces acting on a cross-section consist of four components (force per unit length)the weight of the section (119908) a hydrodynamic lift associatedwith the change of fluid momentum (119891
119872) a viscous lift
force associated with the cross flow drag (119891119888119889) [7 12] and
a buoyancy force associated with instantaneous displacedvolume (119891
119887) [3 12]
51 Momentum and Added Mass Theories The hydrody-namic lift force associated with the change of fluid momen-tum per unit length 119891
119872 acting at a section is [9] as follows
119891119872
=119863
119863119905(119898119886119881) = 119898
119886 + 119881
119886minus
120597
120597120585(119898119886119881)
119889120585
119889119905
= 119898119886 + 119881
119886minus 119880
120597
120597120585(119898119886119881)
(4)
where 119898119886is the added mass associated with the section
form 119880 is the relative fluid velocity parallel to the keel
4 International Journal of Engineering Mathematics
and 119881 is the velocity in plane of the cross-section normal tothe baseline Formula for othermentioned force can be foundin most of previous works [3 4 7 12 13]
In subject of hydrodynamics force port and starboardsides can be considered separately This means that
119891119872PT = minus
119863
119863119905(119898119886PT
119881)
119891119872SB = minus
119863
119863119905(119898119886SB
119881)
(5)
To obtain hydrodynamic force added mass theory will beimplemented Added mass is a widely used concept in avariety of applications like maneuvering seakeeping andplaning calculations The amount of added mass variesaccording to the shape and size of the body The added massfor a V-shaped wedge is given by [3]
119898119886= 119896119886
120587
21205881198872
(6)
and its time derivative is
119889119898119886
119889119905= 119886= 119896119886120587120588119887 (7)
where 119896119886is the added mass coefficient and 119887 is the instanta-
neous half beam of the section Depth of penetration for eachsection is given by [9]
119889 =119887
cot120573 (8)
where 120573 is the deadrise angle Taking into account the effectof water pileup the effective depth of penetration (119889
119890) is
expressed as [9]
119889119890= 119862pu119889 (9)
where119862pu is the pileup or splash-up coefficient Overall it canbe written that [9]
119887 = 119889119890cot120573 = 119862pu119889cot120573 (10)
Hence it can be concluded that the time derivative of theadded mass is
119886= 119896119886120587120588119887 (119862pucot120573) 119889 (11)
Therefore when the immersion exceeds the chine we have[9]
119898119886= 119896119886
120587
21205881198872
max = 119888119905119890
119886= 0
(12)
where 119887max is the half beam at chine Furthermore at anypoint 119875(120585 120577) it can be written [7] as
119909 (119905) = 119909119862119866
(119905) + 120585 cos 120579 (119905) + 120577 sin 120579 (119905)
119911 (119905) = 119911119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905)
(13)
The submergence of a section in terms of the motion will beas follows [7]
ℎ (120585 120577 119905) = 119885119862119866
(119905) minus 120578 (120585 119905)
= 119885119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905) minus 119903 (120585 119905)
(14)
For wavelengths which are long in comparison to the draftand for small wave slopes the immersion of a sectionmeasured perpendicular to the baseline is approximated asin [7]
119889 asympℎ (120585 120577 119905)
cos 120579 (119905) minus V sin 120579 (119905) (15)
where V is the wave slope The rate change of submergence isgiven by [7]
119889 = minus 120578
cos 120579 minus V sin 120579+
119911 minus 120578
(cos 120579 minus V sin 120579)2
120597 (cos 120579 minus ] sin 120579)
120597119905
(16)
Since the immersion 119911 minus 120578 is always small in the valid rangethe relationship can be further simplified to
119889 asymp minus 120578
cos 120579 minus V sin 120579 (17)
Consequently
119886asymp 119896119886120587120588119887 (119862pucot120573)
minus 120578
cos 120579 minus V sin 120579 (18)
52 Total Hydrodynamic Force andMoment The total hydro-dynamic forces acting on the vessel are obtained by inte-grating sectional 2D forces over the wetted length 119897
119908 of the
craft Force and moments in each direction are presentedseparately
521 Horizontal Force The force acting in the horizontal 119909-direction is given by
119865119909= minus (int
119897
119891119872sin 120579119889120585 + int
119897
119891119862119863
sin 120579119889120585) cos120595
= minus (int
119897
(119873119872119875
+ 119873119872119878
) sin 120579119889120585 + int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585)
times cos120595
= minus (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) sin 120579119889120585
+int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585) cos120595
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus 119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
(19)
International Journal of Engineering Mathematics 5
where =
119862119866sin 120579 minus 120579120585 +
119862119866cos 120579 minus
119911cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911
120579 sin 120579
120597119880
120597120585=
120597119908119911
120597120585sin 120579
120597119881
120597120585= minus 120579 minus
120597119908119911
120597120585cos 120579
119889119908119911
119889119905= 119911minus 119880
120597119908119911
120597120585
(20)
By substituting (20) in (19) horizontal force will be as follows
119865119909= minus [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
+ 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875) +
119886119878119881119878
minus119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
= minus [int
119897
119898119886119875119889120585119862119866
sin 120579 minus 120579 int
119897
119898119886119875120585119889120585
+ int
119897
119898119886119875119889120585119862119866
cos 120579 + int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585)
times cos 120579119889120585
+ int
119897
119898119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) 119889120585
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 +int
119897
119886119875119881119875119889120585 +119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
+ Similar S terms ] sin 120579 cos120595
(21)
Now the following definitions will be introduced
119872119886= int
119897
119898119886119889120585
119876119886= int
119897
119898119886120585119889120585
(22)
and subsequently
119865119909= minus [119872
119886119862119866
sin2120579 cos120595 minus 119876119886
120579 sin 120579 cos120595
+ 119872119886119862119866
cos 120579 sin 120579 cos120595
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585 sin 120579 cos120595
+ 119872119886
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) sin 120579 cos120595
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 sin 120579 cos120595
+ int
119897
119886119875119881119875119889120585 sin 120579 cos120595+ 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
sin 120579 cos120595
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 sin 120579 cos120595
+int
119897
1198621198631205881198871198751198812
119875119889120585 sin 120579 cos120595 + Similar S terms]
(23)
522 Side Force Similar to the horizontal force lateral forcecan also be obtained Side force is a result of the differencebetween side force at port and starboard of the craft whichyield sway motion Generally it can be written as follows
119865119910= (int
119897
119891119872119889120585 + int
119897
119891119862119863
119889120585) sin120595 cos 120579
= (int
119897
(119873119872119875
minus 119873119872119878
) 119889120585 plusmn int
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos120579
= (int
119897
(119863
119863119905(119898119886119875119881119875) minus
119863
119863119905(119898119886119878119881119878)) 119889120585
plusmnint
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos 120579
= [int
119897
(119898119886119875119875minus 119898119886119878119878+ 119886119875119881119875minus 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) + 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] sin120595 cos 120579
(24)
Using (20) the side force will be equal to
119865119884
= [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
minus 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875)
minus 119886119878119881119878+ 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] cos 120579 sin120595
(25)
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
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Stochastic AnalysisInternational Journal of
2 International Journal of Engineering Mathematics
published model data for the near-transom pressure Thisreduced the pressure near the stem gradually to zero at thestem vanDayzen [12] extended the originalmodel developedby Zarnick [3 4] and later extended by Keuning [7] to threedegrees of freedom of surge heave and pitch motion inboth regular and irregular head seas The simulations can becarried out with either a constant forward speed or constantthrust He also validated the results by experimental dataof two models and found his model very sensitive to thehull geometry More recently Sebastianii et al [13] developedprevious studies to combine the roll heave and pitch degreesof freedom
In the current paper a mathematical model based on themodels of Zarnick [3] and Sebastianii et al [13] is developedIn all previous studies 2119889 + 119905 theory based symmetric wedgewater entry has been usedHowever the present paper tries todevelop a mathematical model based on asymmetric wedgewater entry which leads to various forces and momentsActually this mathematical model can be considered as afirst step in extending Zarnickrsquos model to six degrees offreedom and the authors are well aware of the fact that thismathematicalmodelmust be graduallymodified step by step
2 Equations of Motions
Planing motions can be divided into two main parts linearand angular motion Moreover various forces and momentsincluding hydrodynamics force hydrostatic force weightand wave effects must be considered in equations of motionHowever no aerodynamic forces are investigated Based onNewtonrsquos second law governing equations of motions can bewritten as
119898 = 119865119909(119905) = 119883
119903(119905) + 119883
119908(119905) + 119883
119891(119905)
119898 119910 = 119865119910(119905) = 119884
119903(119905) + 119884
119908(119905) + 119884
119891(119905)
119898 = 119865119911(119905) = 119885
119903(119905) + 119885
119908(119905) + 119885
119891(119905)
119868119866119909119909
= 119872119909(119905) = 119870
119903(119905) + 119870
119908(119905) + 119870
119891(119905)
119868119866119910119910
120579 = 119872119910(119905) = 119875
119903(119905) + 119875
119908(119905) + 119875
119891(119905)
119868119866119911119911
= 119872119911(119905) = 119877
119903(119905) + 119877
119908(119905) + 119877
119891(119905)
(1)
where subscripts 119903 119908 and 119891 denote hydrostatic hydrody-namic and wave force and moment 119865 and 119872 also indicateforce andmomentwhile119898 119868 and 119905 aremassmoment inertiaand time respectively Equations (1) are written based onthe shown coordinate system To solve equations of motionsforce and moment must be calculated To calculate force andmoment various theories such as 2119889 + 119905 momentum andadded mass theory become necessary
3 2119889 + 119905 Theory
In a ship-fixed coordinate system 25D theory means thatthe two-dimensional equations are solved together withthree-dimensional free surface conditions If the attention is
focused on an Earth-fixed cross-plane one will see a time-dependent problem in 2D cross-plane when the vessel ispassing through it Accordingly the theory is also called 2119889+
119905 theory In an Earth-fixed coordinate system a prismaticplaning vessel of trim angle 120591 (up to 20 degrees) is movingthrough an Earth-fixed cross-plane with speed 119880
119904 as shown
in Figure 1 At time 119905 = 1199050 the cross-section is just above the
free surface at time 119905 = 1199051the cross-section is penetrating
the free surface and at time 119905 = 1199052 flow separates from
the chine line Therefore one can see a process where aV-shaped section enters the water surface in this cross-plane by a speed of 119881 = 119880
119904120591 However it must be noted
that the developed mathematical model has some limitationespecially at high speed Trim angle cannot exceed 20 degreesand the wavelength must also be larger than the hull length
Same procedure can be defined for penetration of anasymmetric wedge into the free surface This means that sideforce can also exist This side force leads to roll momentand yaw moment Therefore various motions of planing hullcan be taken into account Based on this definition twodifferent coordinate systems should be considered (as shownin Figures 2 and 3) Generally three fundamental aspects dueto inclination of section which are shown in Figure 4 can beconsidered [13]
(i) Nonsymmetrical action of fluid on boat the boat inoblique sea undergoes different actions of fluid in itsport and starboard side due to different absolutewavevelocities and relative boat motions relevant to roll
(ii) Nonsymmetry of the section impacting against waterdue to roll motion the section which impacts againstwater is not symmetrical starboard and port sides areconsidered separately with their ldquoequivalent deadriseanglerdquo which is the resultant of the local geometricaldeadrise and the roll angle in order to compute theadded mass terms
(iii) Nonsymmetrical submerged volume geometry sub-merged volume and wet surface are no longer sym-metrical affecting hydrostatics and in general forceapplication points
Finally it is concluded that the force acting on the hull mustbe calculated section by section for starboard side and portside separately
4 Regular Wave Theory
In the present computationalmodel wave forces are obtainedby neglecting diffraction forces (only Froude-Krylov forcesare considered) It is also assumed that the wave excitationis caused by the instantaneous wetted surface and by thevertical component of the wave orbital velocity at the surface119908119911 The influence of the horizontal component of wave
orbital velocity on both the horizontal and vertical motions isneglected because this velocity is considered to be relativelysmall in comparison with the forward speed of the craft Thenormal velocity119881 and the velocity component parallel to thekeel119880 can be written as functions of the craftrsquos forward speed
International Journal of Engineering Mathematics 3
-
Us
t = t0
t = t2
t = t1
V
120591
Earth-fixed cross-plane
Figure 1 Demonstration of 2119889 + 119905 concept [14]
G
BL
WLo
zz998400
Us
x998400
120579
Figure 2 Coordinate system 1 [13]
WL
G
Z
x
Y
120595
120577
o
Figure 3 Coordinate system 2 [13]
heave pitch and vertical component of wave orbital velocity[7] as in
119880 (120577 120585) = 119862119866
(119905) cos 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)] sin 120579 (119905)
119881 (120585 119905) = 119862119866
(119905) sin 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)]
times cos 120579 (119905) minus 120579 (119905) 120585
(2)
FSX
FDX
120573y
z
120573 + 120595
Figure 4 Force acting on an inclined wedge [13]
For regular waves the wave elevation of a linear deep waterwave [13] is
120585 (119909 119910 119905) = 119886 cos (119896119909 cos 120583 + 119896119910 sin 120583 minus 120596119890119905 + 120576) (3)
where 119886 is the wave amplitude 119896 is the wave number and 119888 isthe wave celerity
5 Force Acting on the Hull
As mentioned earlier the numerical model employed herefor prediction of planing motion utilizes a modified 2119889 +
119905 theory with momentum theory The vessel is consideredto be composed of a series of 2D wedges and the threedimensional problem is subsequently solved as a summationof the individual 2D slices The forces acting on a cross-section consist of four components (force per unit length)the weight of the section (119908) a hydrodynamic lift associatedwith the change of fluid momentum (119891
119872) a viscous lift
force associated with the cross flow drag (119891119888119889) [7 12] and
a buoyancy force associated with instantaneous displacedvolume (119891
119887) [3 12]
51 Momentum and Added Mass Theories The hydrody-namic lift force associated with the change of fluid momen-tum per unit length 119891
119872 acting at a section is [9] as follows
119891119872
=119863
119863119905(119898119886119881) = 119898
119886 + 119881
119886minus
120597
120597120585(119898119886119881)
119889120585
119889119905
= 119898119886 + 119881
119886minus 119880
120597
120597120585(119898119886119881)
(4)
where 119898119886is the added mass associated with the section
form 119880 is the relative fluid velocity parallel to the keel
4 International Journal of Engineering Mathematics
and 119881 is the velocity in plane of the cross-section normal tothe baseline Formula for othermentioned force can be foundin most of previous works [3 4 7 12 13]
In subject of hydrodynamics force port and starboardsides can be considered separately This means that
119891119872PT = minus
119863
119863119905(119898119886PT
119881)
119891119872SB = minus
119863
119863119905(119898119886SB
119881)
(5)
To obtain hydrodynamic force added mass theory will beimplemented Added mass is a widely used concept in avariety of applications like maneuvering seakeeping andplaning calculations The amount of added mass variesaccording to the shape and size of the body The added massfor a V-shaped wedge is given by [3]
119898119886= 119896119886
120587
21205881198872
(6)
and its time derivative is
119889119898119886
119889119905= 119886= 119896119886120587120588119887 (7)
where 119896119886is the added mass coefficient and 119887 is the instanta-
neous half beam of the section Depth of penetration for eachsection is given by [9]
119889 =119887
cot120573 (8)
where 120573 is the deadrise angle Taking into account the effectof water pileup the effective depth of penetration (119889
119890) is
expressed as [9]
119889119890= 119862pu119889 (9)
where119862pu is the pileup or splash-up coefficient Overall it canbe written that [9]
119887 = 119889119890cot120573 = 119862pu119889cot120573 (10)
Hence it can be concluded that the time derivative of theadded mass is
119886= 119896119886120587120588119887 (119862pucot120573) 119889 (11)
Therefore when the immersion exceeds the chine we have[9]
119898119886= 119896119886
120587
21205881198872
max = 119888119905119890
119886= 0
(12)
where 119887max is the half beam at chine Furthermore at anypoint 119875(120585 120577) it can be written [7] as
119909 (119905) = 119909119862119866
(119905) + 120585 cos 120579 (119905) + 120577 sin 120579 (119905)
119911 (119905) = 119911119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905)
(13)
The submergence of a section in terms of the motion will beas follows [7]
ℎ (120585 120577 119905) = 119885119862119866
(119905) minus 120578 (120585 119905)
= 119885119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905) minus 119903 (120585 119905)
(14)
For wavelengths which are long in comparison to the draftand for small wave slopes the immersion of a sectionmeasured perpendicular to the baseline is approximated asin [7]
119889 asympℎ (120585 120577 119905)
cos 120579 (119905) minus V sin 120579 (119905) (15)
where V is the wave slope The rate change of submergence isgiven by [7]
119889 = minus 120578
cos 120579 minus V sin 120579+
119911 minus 120578
(cos 120579 minus V sin 120579)2
120597 (cos 120579 minus ] sin 120579)
120597119905
(16)
Since the immersion 119911 minus 120578 is always small in the valid rangethe relationship can be further simplified to
119889 asymp minus 120578
cos 120579 minus V sin 120579 (17)
Consequently
119886asymp 119896119886120587120588119887 (119862pucot120573)
minus 120578
cos 120579 minus V sin 120579 (18)
52 Total Hydrodynamic Force andMoment The total hydro-dynamic forces acting on the vessel are obtained by inte-grating sectional 2D forces over the wetted length 119897
119908 of the
craft Force and moments in each direction are presentedseparately
521 Horizontal Force The force acting in the horizontal 119909-direction is given by
119865119909= minus (int
119897
119891119872sin 120579119889120585 + int
119897
119891119862119863
sin 120579119889120585) cos120595
= minus (int
119897
(119873119872119875
+ 119873119872119878
) sin 120579119889120585 + int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585)
times cos120595
= minus (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) sin 120579119889120585
+int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585) cos120595
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus 119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
(19)
International Journal of Engineering Mathematics 5
where =
119862119866sin 120579 minus 120579120585 +
119862119866cos 120579 minus
119911cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911
120579 sin 120579
120597119880
120597120585=
120597119908119911
120597120585sin 120579
120597119881
120597120585= minus 120579 minus
120597119908119911
120597120585cos 120579
119889119908119911
119889119905= 119911minus 119880
120597119908119911
120597120585
(20)
By substituting (20) in (19) horizontal force will be as follows
119865119909= minus [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
+ 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875) +
119886119878119881119878
minus119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
= minus [int
119897
119898119886119875119889120585119862119866
sin 120579 minus 120579 int
119897
119898119886119875120585119889120585
+ int
119897
119898119886119875119889120585119862119866
cos 120579 + int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585)
times cos 120579119889120585
+ int
119897
119898119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) 119889120585
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 +int
119897
119886119875119881119875119889120585 +119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
+ Similar S terms ] sin 120579 cos120595
(21)
Now the following definitions will be introduced
119872119886= int
119897
119898119886119889120585
119876119886= int
119897
119898119886120585119889120585
(22)
and subsequently
119865119909= minus [119872
119886119862119866
sin2120579 cos120595 minus 119876119886
120579 sin 120579 cos120595
+ 119872119886119862119866
cos 120579 sin 120579 cos120595
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585 sin 120579 cos120595
+ 119872119886
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) sin 120579 cos120595
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 sin 120579 cos120595
+ int
119897
119886119875119881119875119889120585 sin 120579 cos120595+ 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
sin 120579 cos120595
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 sin 120579 cos120595
+int
119897
1198621198631205881198871198751198812
119875119889120585 sin 120579 cos120595 + Similar S terms]
(23)
522 Side Force Similar to the horizontal force lateral forcecan also be obtained Side force is a result of the differencebetween side force at port and starboard of the craft whichyield sway motion Generally it can be written as follows
119865119910= (int
119897
119891119872119889120585 + int
119897
119891119862119863
119889120585) sin120595 cos 120579
= (int
119897
(119873119872119875
minus 119873119872119878
) 119889120585 plusmn int
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos120579
= (int
119897
(119863
119863119905(119898119886119875119881119875) minus
119863
119863119905(119898119886119878119881119878)) 119889120585
plusmnint
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos 120579
= [int
119897
(119898119886119875119875minus 119898119886119878119878+ 119886119875119881119875minus 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) + 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] sin120595 cos 120579
(24)
Using (20) the side force will be equal to
119865119884
= [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
minus 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875)
minus 119886119878119881119878+ 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] cos 120579 sin120595
(25)
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 3
-
Us
t = t0
t = t2
t = t1
V
120591
Earth-fixed cross-plane
Figure 1 Demonstration of 2119889 + 119905 concept [14]
G
BL
WLo
zz998400
Us
x998400
120579
Figure 2 Coordinate system 1 [13]
WL
G
Z
x
Y
120595
120577
o
Figure 3 Coordinate system 2 [13]
heave pitch and vertical component of wave orbital velocity[7] as in
119880 (120577 120585) = 119862119866
(119905) cos 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)] sin 120579 (119905)
119881 (120585 119905) = 119862119866
(119905) sin 120579 (119905) minus [119862119866
(119905) minus 119908119911(120585 119905)]
times cos 120579 (119905) minus 120579 (119905) 120585
(2)
FSX
FDX
120573y
z
120573 + 120595
Figure 4 Force acting on an inclined wedge [13]
For regular waves the wave elevation of a linear deep waterwave [13] is
120585 (119909 119910 119905) = 119886 cos (119896119909 cos 120583 + 119896119910 sin 120583 minus 120596119890119905 + 120576) (3)
where 119886 is the wave amplitude 119896 is the wave number and 119888 isthe wave celerity
5 Force Acting on the Hull
As mentioned earlier the numerical model employed herefor prediction of planing motion utilizes a modified 2119889 +
119905 theory with momentum theory The vessel is consideredto be composed of a series of 2D wedges and the threedimensional problem is subsequently solved as a summationof the individual 2D slices The forces acting on a cross-section consist of four components (force per unit length)the weight of the section (119908) a hydrodynamic lift associatedwith the change of fluid momentum (119891
119872) a viscous lift
force associated with the cross flow drag (119891119888119889) [7 12] and
a buoyancy force associated with instantaneous displacedvolume (119891
119887) [3 12]
51 Momentum and Added Mass Theories The hydrody-namic lift force associated with the change of fluid momen-tum per unit length 119891
119872 acting at a section is [9] as follows
119891119872
=119863
119863119905(119898119886119881) = 119898
119886 + 119881
119886minus
120597
120597120585(119898119886119881)
119889120585
119889119905
= 119898119886 + 119881
119886minus 119880
120597
120597120585(119898119886119881)
(4)
where 119898119886is the added mass associated with the section
form 119880 is the relative fluid velocity parallel to the keel
4 International Journal of Engineering Mathematics
and 119881 is the velocity in plane of the cross-section normal tothe baseline Formula for othermentioned force can be foundin most of previous works [3 4 7 12 13]
In subject of hydrodynamics force port and starboardsides can be considered separately This means that
119891119872PT = minus
119863
119863119905(119898119886PT
119881)
119891119872SB = minus
119863
119863119905(119898119886SB
119881)
(5)
To obtain hydrodynamic force added mass theory will beimplemented Added mass is a widely used concept in avariety of applications like maneuvering seakeeping andplaning calculations The amount of added mass variesaccording to the shape and size of the body The added massfor a V-shaped wedge is given by [3]
119898119886= 119896119886
120587
21205881198872
(6)
and its time derivative is
119889119898119886
119889119905= 119886= 119896119886120587120588119887 (7)
where 119896119886is the added mass coefficient and 119887 is the instanta-
neous half beam of the section Depth of penetration for eachsection is given by [9]
119889 =119887
cot120573 (8)
where 120573 is the deadrise angle Taking into account the effectof water pileup the effective depth of penetration (119889
119890) is
expressed as [9]
119889119890= 119862pu119889 (9)
where119862pu is the pileup or splash-up coefficient Overall it canbe written that [9]
119887 = 119889119890cot120573 = 119862pu119889cot120573 (10)
Hence it can be concluded that the time derivative of theadded mass is
119886= 119896119886120587120588119887 (119862pucot120573) 119889 (11)
Therefore when the immersion exceeds the chine we have[9]
119898119886= 119896119886
120587
21205881198872
max = 119888119905119890
119886= 0
(12)
where 119887max is the half beam at chine Furthermore at anypoint 119875(120585 120577) it can be written [7] as
119909 (119905) = 119909119862119866
(119905) + 120585 cos 120579 (119905) + 120577 sin 120579 (119905)
119911 (119905) = 119911119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905)
(13)
The submergence of a section in terms of the motion will beas follows [7]
ℎ (120585 120577 119905) = 119885119862119866
(119905) minus 120578 (120585 119905)
= 119885119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905) minus 119903 (120585 119905)
(14)
For wavelengths which are long in comparison to the draftand for small wave slopes the immersion of a sectionmeasured perpendicular to the baseline is approximated asin [7]
119889 asympℎ (120585 120577 119905)
cos 120579 (119905) minus V sin 120579 (119905) (15)
where V is the wave slope The rate change of submergence isgiven by [7]
119889 = minus 120578
cos 120579 minus V sin 120579+
119911 minus 120578
(cos 120579 minus V sin 120579)2
120597 (cos 120579 minus ] sin 120579)
120597119905
(16)
Since the immersion 119911 minus 120578 is always small in the valid rangethe relationship can be further simplified to
119889 asymp minus 120578
cos 120579 minus V sin 120579 (17)
Consequently
119886asymp 119896119886120587120588119887 (119862pucot120573)
minus 120578
cos 120579 minus V sin 120579 (18)
52 Total Hydrodynamic Force andMoment The total hydro-dynamic forces acting on the vessel are obtained by inte-grating sectional 2D forces over the wetted length 119897
119908 of the
craft Force and moments in each direction are presentedseparately
521 Horizontal Force The force acting in the horizontal 119909-direction is given by
119865119909= minus (int
119897
119891119872sin 120579119889120585 + int
119897
119891119862119863
sin 120579119889120585) cos120595
= minus (int
119897
(119873119872119875
+ 119873119872119878
) sin 120579119889120585 + int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585)
times cos120595
= minus (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) sin 120579119889120585
+int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585) cos120595
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus 119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
(19)
International Journal of Engineering Mathematics 5
where =
119862119866sin 120579 minus 120579120585 +
119862119866cos 120579 minus
119911cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911
120579 sin 120579
120597119880
120597120585=
120597119908119911
120597120585sin 120579
120597119881
120597120585= minus 120579 minus
120597119908119911
120597120585cos 120579
119889119908119911
119889119905= 119911minus 119880
120597119908119911
120597120585
(20)
By substituting (20) in (19) horizontal force will be as follows
119865119909= minus [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
+ 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875) +
119886119878119881119878
minus119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
= minus [int
119897
119898119886119875119889120585119862119866
sin 120579 minus 120579 int
119897
119898119886119875120585119889120585
+ int
119897
119898119886119875119889120585119862119866
cos 120579 + int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585)
times cos 120579119889120585
+ int
119897
119898119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) 119889120585
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 +int
119897
119886119875119881119875119889120585 +119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
+ Similar S terms ] sin 120579 cos120595
(21)
Now the following definitions will be introduced
119872119886= int
119897
119898119886119889120585
119876119886= int
119897
119898119886120585119889120585
(22)
and subsequently
119865119909= minus [119872
119886119862119866
sin2120579 cos120595 minus 119876119886
120579 sin 120579 cos120595
+ 119872119886119862119866
cos 120579 sin 120579 cos120595
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585 sin 120579 cos120595
+ 119872119886
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) sin 120579 cos120595
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 sin 120579 cos120595
+ int
119897
119886119875119881119875119889120585 sin 120579 cos120595+ 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
sin 120579 cos120595
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 sin 120579 cos120595
+int
119897
1198621198631205881198871198751198812
119875119889120585 sin 120579 cos120595 + Similar S terms]
(23)
522 Side Force Similar to the horizontal force lateral forcecan also be obtained Side force is a result of the differencebetween side force at port and starboard of the craft whichyield sway motion Generally it can be written as follows
119865119910= (int
119897
119891119872119889120585 + int
119897
119891119862119863
119889120585) sin120595 cos 120579
= (int
119897
(119873119872119875
minus 119873119872119878
) 119889120585 plusmn int
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos120579
= (int
119897
(119863
119863119905(119898119886119875119881119875) minus
119863
119863119905(119898119886119878119881119878)) 119889120585
plusmnint
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos 120579
= [int
119897
(119898119886119875119875minus 119898119886119878119878+ 119886119875119881119875minus 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) + 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] sin120595 cos 120579
(24)
Using (20) the side force will be equal to
119865119884
= [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
minus 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875)
minus 119886119878119881119878+ 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] cos 120579 sin120595
(25)
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
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Stochastic AnalysisInternational Journal of
4 International Journal of Engineering Mathematics
and 119881 is the velocity in plane of the cross-section normal tothe baseline Formula for othermentioned force can be foundin most of previous works [3 4 7 12 13]
In subject of hydrodynamics force port and starboardsides can be considered separately This means that
119891119872PT = minus
119863
119863119905(119898119886PT
119881)
119891119872SB = minus
119863
119863119905(119898119886SB
119881)
(5)
To obtain hydrodynamic force added mass theory will beimplemented Added mass is a widely used concept in avariety of applications like maneuvering seakeeping andplaning calculations The amount of added mass variesaccording to the shape and size of the body The added massfor a V-shaped wedge is given by [3]
119898119886= 119896119886
120587
21205881198872
(6)
and its time derivative is
119889119898119886
119889119905= 119886= 119896119886120587120588119887 (7)
where 119896119886is the added mass coefficient and 119887 is the instanta-
neous half beam of the section Depth of penetration for eachsection is given by [9]
119889 =119887
cot120573 (8)
where 120573 is the deadrise angle Taking into account the effectof water pileup the effective depth of penetration (119889
119890) is
expressed as [9]
119889119890= 119862pu119889 (9)
where119862pu is the pileup or splash-up coefficient Overall it canbe written that [9]
119887 = 119889119890cot120573 = 119862pu119889cot120573 (10)
Hence it can be concluded that the time derivative of theadded mass is
119886= 119896119886120587120588119887 (119862pucot120573) 119889 (11)
Therefore when the immersion exceeds the chine we have[9]
119898119886= 119896119886
120587
21205881198872
max = 119888119905119890
119886= 0
(12)
where 119887max is the half beam at chine Furthermore at anypoint 119875(120585 120577) it can be written [7] as
119909 (119905) = 119909119862119866
(119905) + 120585 cos 120579 (119905) + 120577 sin 120579 (119905)
119911 (119905) = 119911119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905)
(13)
The submergence of a section in terms of the motion will beas follows [7]
ℎ (120585 120577 119905) = 119885119862119866
(119905) minus 120578 (120585 119905)
= 119885119862119866
(119905) minus 120585 sin 120579 (119905) + 120577 cos 120579 (119905) minus 119903 (120585 119905)
(14)
For wavelengths which are long in comparison to the draftand for small wave slopes the immersion of a sectionmeasured perpendicular to the baseline is approximated asin [7]
119889 asympℎ (120585 120577 119905)
cos 120579 (119905) minus V sin 120579 (119905) (15)
where V is the wave slope The rate change of submergence isgiven by [7]
119889 = minus 120578
cos 120579 minus V sin 120579+
119911 minus 120578
(cos 120579 minus V sin 120579)2
120597 (cos 120579 minus ] sin 120579)
120597119905
(16)
Since the immersion 119911 minus 120578 is always small in the valid rangethe relationship can be further simplified to
119889 asymp minus 120578
cos 120579 minus V sin 120579 (17)
Consequently
119886asymp 119896119886120587120588119887 (119862pucot120573)
minus 120578
cos 120579 minus V sin 120579 (18)
52 Total Hydrodynamic Force andMoment The total hydro-dynamic forces acting on the vessel are obtained by inte-grating sectional 2D forces over the wetted length 119897
119908 of the
craft Force and moments in each direction are presentedseparately
521 Horizontal Force The force acting in the horizontal 119909-direction is given by
119865119909= minus (int
119897
119891119872sin 120579119889120585 + int
119897
119891119862119863
sin 120579119889120585) cos120595
= minus (int
119897
(119873119872119875
+ 119873119872119878
) sin 120579119889120585 + int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585)
times cos120595
= minus (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) sin 120579119889120585
+int
119897
(119891119862119863119875
+ 119891119862119863119878
) sin 120579119889120585) cos120595
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus 119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
(19)
International Journal of Engineering Mathematics 5
where =
119862119866sin 120579 minus 120579120585 +
119862119866cos 120579 minus
119911cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911
120579 sin 120579
120597119880
120597120585=
120597119908119911
120597120585sin 120579
120597119881
120597120585= minus 120579 minus
120597119908119911
120597120585cos 120579
119889119908119911
119889119905= 119911minus 119880
120597119908119911
120597120585
(20)
By substituting (20) in (19) horizontal force will be as follows
119865119909= minus [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
+ 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875) +
119886119878119881119878
minus119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
= minus [int
119897
119898119886119875119889120585119862119866
sin 120579 minus 120579 int
119897
119898119886119875120585119889120585
+ int
119897
119898119886119875119889120585119862119866
cos 120579 + int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585)
times cos 120579119889120585
+ int
119897
119898119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) 119889120585
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 +int
119897
119886119875119881119875119889120585 +119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
+ Similar S terms ] sin 120579 cos120595
(21)
Now the following definitions will be introduced
119872119886= int
119897
119898119886119889120585
119876119886= int
119897
119898119886120585119889120585
(22)
and subsequently
119865119909= minus [119872
119886119862119866
sin2120579 cos120595 minus 119876119886
120579 sin 120579 cos120595
+ 119872119886119862119866
cos 120579 sin 120579 cos120595
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585 sin 120579 cos120595
+ 119872119886
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) sin 120579 cos120595
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 sin 120579 cos120595
+ int
119897
119886119875119881119875119889120585 sin 120579 cos120595+ 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
sin 120579 cos120595
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 sin 120579 cos120595
+int
119897
1198621198631205881198871198751198812
119875119889120585 sin 120579 cos120595 + Similar S terms]
(23)
522 Side Force Similar to the horizontal force lateral forcecan also be obtained Side force is a result of the differencebetween side force at port and starboard of the craft whichyield sway motion Generally it can be written as follows
119865119910= (int
119897
119891119872119889120585 + int
119897
119891119862119863
119889120585) sin120595 cos 120579
= (int
119897
(119873119872119875
minus 119873119872119878
) 119889120585 plusmn int
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos120579
= (int
119897
(119863
119863119905(119898119886119875119881119875) minus
119863
119863119905(119898119886119878119881119878)) 119889120585
plusmnint
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos 120579
= [int
119897
(119898119886119875119875minus 119898119886119878119878+ 119886119875119881119875minus 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) + 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] sin120595 cos 120579
(24)
Using (20) the side force will be equal to
119865119884
= [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
minus 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875)
minus 119886119878119881119878+ 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] cos 120579 sin120595
(25)
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
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Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 5
where =
119862119866sin 120579 minus 120579120585 +
119862119866cos 120579 minus
119911cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911
120579 sin 120579
120597119880
120597120585=
120597119908119911
120597120585sin 120579
120597119881
120597120585= minus 120579 minus
120597119908119911
120597120585cos 120579
119889119908119911
119889119905= 119911minus 119880
120597119908119911
120597120585
(20)
By substituting (20) in (19) horizontal force will be as follows
119865119909= minus [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
+ 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875) +
119886119878119881119878
minus119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] sin 120579 cos120595
= minus [int
119897
119898119886119875119889120585119862119866
sin 120579 minus 120579 int
119897
119898119886119875120585119889120585
+ int
119897
119898119886119875119889120585119862119866
cos 120579 + int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585)
times cos 120579119889120585
+ int
119897
119898119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) 119889120585
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 +int
119897
119886119875119881119875119889120585 +119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
+ Similar S terms ] sin 120579 cos120595
(21)
Now the following definitions will be introduced
119872119886= int
119897
119898119886119889120585
119876119886= int
119897
119898119886120585119889120585
(22)
and subsequently
119865119909= minus [119872
119886119862119866
sin2120579 cos120595 minus 119876119886
120579 sin 120579 cos120595
+ 119872119886119862119866
cos 120579 sin 120579 cos120595
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585 sin 120579 cos120595
+ 119872119886
120579 (119862119866
cos 120579 minus 119862119866
sin 120579) sin 120579 cos120595
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 sin 120579 cos120595
+ int
119897
119886119875119881119875119889120585 sin 120579 cos120595+ 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
sin 120579 cos120595
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 sin 120579 cos120595
+int
119897
1198621198631205881198871198751198812
119875119889120585 sin 120579 cos120595 + Similar S terms]
(23)
522 Side Force Similar to the horizontal force lateral forcecan also be obtained Side force is a result of the differencebetween side force at port and starboard of the craft whichyield sway motion Generally it can be written as follows
119865119910= (int
119897
119891119872119889120585 + int
119897
119891119862119863
119889120585) sin120595 cos 120579
= (int
119897
(119873119872119875
minus 119873119872119878
) 119889120585 plusmn int
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos120579
= (int
119897
(119863
119863119905(119898119886119875119881119875) minus
119863
119863119905(119898119886119878119881119878)) 119889120585
plusmnint
119897
(119891119862119863119875
minus 119891119862119863119878
) 119889120585) sin120595 cos 120579
= [int
119897
(119898119886119875119875minus 119898119886119878119878+ 119886119875119881119875minus 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) + 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] sin120595 cos 120579
(24)
Using (20) the side force will be equal to
119865119884
= [int
119897
(119898119886119875
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119875cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119875
120579 sin 120579)
minus 119898119886119878
(119862119866
sin 120579 minus 120579120585 + 119862119866
cos 120579 minus 119911119878cos 120579
+ 120579 (119862119866
cos 120579 minus 119862119866
sin 120579) + 119908119911119878
120579 sin 120579)
+ 119886119875119881119875minus 119880119875
120597
120597120585(119898119886119875119881119875)
minus 119886119878119881119878+ 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875minus 1198871198781198812
119878)) 119889120585] cos 120579 sin120595
(25)
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
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Differential EquationsInternational Journal of
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
6 International Journal of Engineering Mathematics
Finally one can write
119865119910= [119872
119886119875119862119866
sin 120579 minus 120579119876119886119875
+ 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
+ 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 + int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585 + int
119897
1198621198631205881198871198751198812
119875119889120585
minus Similar S terms ] cos 120579 sin120595
(26)
523 Vertical Force Same as other forces vertical force willbe as follows
119865119911= (int
119897
119891119872119889120585 minus int
119897
119891119862119863
119889120585) cos120595 cos 120579 minus int
119897
119891119861119889120585
= (int
119897
(119873119872119875
+ 119873119872119878
) 119889120585 minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585)
times cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= (int
119897
(119863
119863119905(119898119886119875119881119875) +
119863
119863119905(119898119886119878119881119878)) 119889120585
minus int
119897
(119891119862119863119875
+ 119891119862119863119878
) 119889120585) cos120595 cos 120579 minus int
119897
119886119861119865
120588119892119860119889120585
= minus [int
119897
(119898119886119875119875+ 119898119886119878119878+ 119886119875119881119875+ 119886119878119881119878
minus119880119875
120597
120597120585(119898119886119875119881119875) minus 119880119878
120597
120597120585(119898119886119878119881119878)) 119889120585
+int
119897
(119862119863120588 (1198871198751198812
119875+ 1198871198781198812
119878)) 119889120585] cos120595 cos 120579
minus int
119897
119886119861119865
120588119892119860119889120585
(27)
Again using (20) we have
119865119911= [ minus 119872
119886119875119862119866
sin 120579 + 119876119886119875
120579 minus 119872119886119875119862119866
cos 120579
+ int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579119889120585
minus 119872119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
minus int
119897
119898119886119875119908119911119875
120579 sin 120579119889120585 minus int
119897
119886119875119881119875119889120585 + 119880
119875119881119875119898119886119875
10038161003816100381610038161003816
sternbow
minus int
119897
119898119886119875119881119875
120597119880119875
120597120585119889120585
minusint
119897
1198621198631205881198871198751198812
119875119889120585 + Similar S terms]
times cos 120579 sin120595 minus int
119897
119886119861119865
120588119892119860119889120585
(28)
524 Roll Moment When all the hydrodynamics forcesare determined it can be an easy task to compute variousmoments acting on the hull Roll moment (119872
119909) is due to side
and vertical forces which can be considered as follows
119872119909= 119865119910119911119888+ 119865119911119910119888 (29)
where 119911119888and 119910
119888are the distance from 119862119866 to center of action
for side and vertical forces successively which can easily becalculated
525 Pitch Moment Pitch moment can also be computedsimilar to roll moment However there exist two ways forpitch moment calculation In the first method it is enoughto act similar to the roll moment This means that
119872119910= 119865119911119909119888 (30)
where 119909119888is the horizontal distance from 119862119866 to center of
action for vertical force In the second method we canintegrate sectional 2D moments over the wetted length 119868 ofthe craft as follows
119872119910= [119876
119886119875119862119866
sin 120579 minus 119868119886119875
120579 + 119876119886119875119862119866
cos 120579
minus int
119897
119898119886119875
(
119889119908119911119875
119889119905+ 119880
120597119908119911119875
120597120585) cos 120579120585119889120585
minus 119876119886119875
120579 (119862119866
cos 120579 minus 119862119866
sin 120579)
+ int
119897
119898119886119875119908119911119875
120579 sin 120579120585119889120585 + int
119897
119886119875119881119875120585119889120585
+ 11988011987511988111987511989811988611987512058510038161003816100381610038161003816stern
+ int
119897
119898119886119875119881119875
120597119908119911119875
120597120585sin 120579120585119889120585
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 7
+int
119897
1198621198631205881198871198751198812
119875120585119889120585 + Similar S terms] sdot cos120595
minus int
119897
119886119861119865
120588119892119860 cos 120579120585119889120585
(31)
526 Yaw Moment Yaw moment is as follows
119872119911= 119865119910119909119888 (32)
where 119909119888is the horizontal distance from 119862119866 to center of
action for the side force Now equations of motion can besolved to determine the time domain motions of the planinghull
6 Solution of Equations of Motion
The solution of the derived equations of motion is com-plicated They form a set of three coupled second-ordernonlinear differential equations which has to be solved usingstandard numerical techniques in the time domain The setof equations is first transformed into a set of six coupledfirst-order nonlinear differential equations by introducing astate vector Subsequently resulting equations must be solvedusing a numerical method such as Runge-Kutta-Merson
Knowing the initial state variables at time instant 1199050
the equations are simultaneously solved for the small timeincrement Δ119905 to yield the solution at 119905 + Δ119905 The advantage ofthe Runge-Kutta-Merson method is that it is high order andit has adaptive step size control More details can be found inmany reference books like [15]
7 Validation
It must be denoted that based on our knowledge there isno experimental or numerical work on planing motion in6 degrees of freedom Therefore to validate the developedmathematical model it is reasonable to examine the basisof the developed code For this purpose experiments ofFridsma [5] are considered He used a prismatic hull with10 20 and 30 deadrise angles in his experiments (Figure 5)Moreover characteristics of the hull which is considered inthe current study are presented in Table 1 To validate thecurrent solutions planing motion at both calm water andregular wave will be compared against the experiments Atfirst resistance of the ship hull at calm water is obtainedand compared against experiments of Fridsma and thenplaning motion at regular wave for eighteen different casesis investigated at different wavelength and wave height whichare presented in Table 2
In addition to some details like the designated parametersin Table 2 more details should be considered to performsimulations For example ship hull is divided into 76 sectionsand initial conditions are adopted based on [3 5] Figure 6indicates that numerical details which are adopted in 2119889 +
119905 simulations are completely in good agreement with thephysical characteristics of the problem In fact the obtainedresistance from 2119889+119905 solutions is in excellent agreement with
Table 1 Characteristics of the considered prismatic hull
Model A B119871119887 5 5120573 (deg) 20 20LCG (119871) 59 62120591 (deg) 4 4119881radic119871 4 6119862Δ
0608 0608
Table 2 Wave characteristics for eighteen different cases
Run conditionsNo 119867119887 120582119871 ℎ1198671 0111 1 0182 0167 1 0173 0222 1 0174 0056 2 0935 0111 2 0846 0167 2 0817 0167 2 0748 0222 2 0739 0334 2 05910 0334 2 06111 0111 3 11812 0222 3 11313 0334 3 10514 0111 4 12315 0222 4 11116 0334 4 10717 0111 6 10418 0334 6 097
the experimental dataTherefore it can be concluded that theconsidered setting may be suitable for future regular wavesolutions
Furthermore Figure 7 shows the obtained results for theheave and pitch motions at different 119867119861 ratios Details ofthe considered variables are reported in Table 2 It is observedthat for wavelength equal to the ship length the obtainedresults are not accurate and an over prediction is seenHowever by increasing the wavelength the results are moreaccurate This can be attributed to the assumption that thewavelength must be sufficiently larger than the ship lengthThis assumption has been utilized in all previous studiesOverall it can be concluded that the current mathematicalmodel can be implemented for practical design of planinghulls seakeeping However it is worth mentioning that thereis urgency for measuring planing craft motions (6-DOF) atregular and irregular waves as a benchmark case
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 International Journal of Engineering Mathematics
Keel
Chin
e 30
Chin
e 20
Chin
e 10
Y
x
9998400998400
302010
Figure 5 Prismatic hull
0
005
01
015
02
025
03
0 1 2 3 4 5 6
RW
Fn
2d + t
EXP
Figure 6 Hull resistance at calm water
8 Results
After validation it is necessary to study the planing craftmotion in six degrees of freedom in regular waves For thispurpose cases 15 and 17 in Table 2 are considered Ship hullis divided to 76 sections and initial condition for each degreeof freedom is arbitrarily adopted These initial values canbe identified using the presented results Moreover threeinitial roll angles 0 5 and 10 degrees are compared againsteach other This means that 119867 and 120582 are kept fixed (cases15 and 17) and effects of an initial roll angle on planingcraft motion are studied Initial yaw and sway values are alsokept to be zero It must be mentioned that the main purposeof these simulations is the examination of the developedmathematical model
The obtained results are shown in Figures 8 and 9Figure 8 shows the results related to case 15 First roll angleis set to zero No sway or yaw motion occurs This is due tothe fact that there is no asymmetric force which can lead toyaw moment and sway motion Therefore heave and pitchmotions and vertical acceleration will be regular
Afterward an initial roll equal to 5 degrees is examinedInitial conditions are the same as in the previous case exceptfor the roll angle It is observed that the roll angle is dampedafter 8 seconds and again increased This leads to irregularbehavior of heave and pitch in regular wave condition Yawand sway of planing hull are also increased by time Theseresults can be analyzed by the fact that damping force actson the roll motion and decreases it However due to theasymmetric fluid flow in roll motion some yaw momentand sway force are generated and regular wave force causesa severe increase in sway and yaw motions Consequentlyroll motion will also be intensified Due to these behaviorsirregular heave and pitch motions exist
When roll angle is increased to 10 degrees a similarbehavior can be seen too However magnitudes of planingmotions are different It is observed that roll motion isrelatively damped at 3 until 5 seconds At the same timeheave and pitch motions remain constant and consequentlyvertical accelerations due to water impact phenomenon areomitted However encounter wave acts on the hull and leadsto a new roll angle In the meantime yaw and sway motionsincrease at a relatively constant rate
In addition to case 15 case 17 (Table 2) is also consideredwith the same methodology Initial roll angle is defined andit is observed that at zero roll angle planing hull has a regularbehavior It is clearly seen that no sway and yaw motions arecreated and that the presented mathematical model worksappropriately As expected bow acceleration is also largerthan the119862119866 acceleration It is due to this fact that main waterentry phenomenon occurs at the fore part of the hull
At 5 degrees roll angle after 8 seconds roll is damped andheave of the hull is increased and consequently pitchmotionis relatively damped However roll motion is affected by theencounter wave and is thus intensified Moreover due to thewave effects and unsteady rollmotion sway and yaw continueby a constant rate Finally it must bementioned that variationof surge velocity is not yet completely modeled and must beconsidered in the next version of the developed code
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 9
0
05
1
15
2
25
0 01 02 03
120582L = 1 exp120582L = 1 code
0
001
002
003
004
0 02 04
120582L = 1 exp120582L = 1 code
HB HB
Pitc
h (d
eg)
h
(a)
0
005
01
015
02
02 040
120582L = 2 exp120582L = 2 code
HB HB
0
1
2
3
4
5
6
0 02 04
120582L = 2 exp120582L = 2 code
Pitc
h (d
eg)
h
(b)
0
2
4
6
8
0 02 04
120582L = 3 exp120582L = 3 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 3 exp120582L = 3 code
HB HB
Pitc
h (d
eg)
h
(c)
Figure 7 Continued
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 International Journal of Engineering Mathematics
0
1
2
3
4
5
6
0 01 02 03 04
120582L = 4 exp120582L = 4 code
0
005
01
015
02
025
03
0 01 02 03 04
120582L = 4120582L = 4
HB HB
Pitc
h (d
eg)
h
(d)
0
1
2
3
4
0 02 040
005
01
015
02
025
03
0 02 04
120582L = 6 exp120582L = 6 code
120582L = 6 exp120582L = 6 code
HB HB
Pitc
h (d
eg)
h
(e)
Figure 7 Comparison of heave and pitch motions with experiments of Fridsma [5]
In the last part of case 17 roll angle of 10 degrees isconsidered Similar behavior relative to the third part of case15 can be observed In a range of time roll is damped andas a result heave pitch and acceleration become constantFurther studies must be performed for understanding thesephysics
9 Conclusions
In this paper various theories ofmomentum addedmass and2119889 + 119905 theories are implemented to develop a mathematicalmodel for simulation of six degrees of freedom motion ofa planing craft in regular waves Therefore 2119889 + 119905 theory isdeveloped for asymmetric wedge water entry and a set offormulas is derived for computation of various forces and
moments on planing hulls Solution of equations of motionsis also considered by awell-knownnumericalmethodRunge-Kutta-Merson which controls the time step size efficiently
In the absence of any six degrees computational dataor experiments for planing craft motions it was decided tovalidate the present model by using Fridsmarsquos experimentin regular waves for heave and pitch motions Comparisonsindicate that the developed code can model planing motionreasonably accurate Furthermore to demonstrate the modelcapability for six degrees of freedom computations Fridsmamodel is considered and effort was made to study theplaning hull behavior at the initial roll angles Three rollangles including 0 5 and 10 degrees are considered and thebehavior of the planing hull is studied It is observed thatfor different wave conditions similar behavior is observed
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 11
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
1
(deg
)
Roll
minus10 2 4 6 8 10 12
(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
0
1
(deg
)
Yaw
minus10 2 4 6 8 10 12
(s)
0
1
Met
er
Sway
minus10 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
(s)0 2 4 6 8 10 12
(a) Zero degree roll angle
0 2 4 6 8 10 120
02
04
(s)
Met
er
Sway
0
20
40
(deg
)
Yaw
0 2 4 6 8 10 12(s)
0
02
G
CG acc
minus020 2 4 6 8 10 12
(s)
4
6
8
(deg
)
Pitch
0 2 4 6 8 10 12(s)
0
005
01
Met
er
Heave
0 2 4 6 8 10 12(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
(b) Five degrees roll angle
Figure 8 Continued
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 International Journal of Engineering Mathematics
0 2 4 6 8 10 120
005
01
(s)
Met
erHeave
0
5
10Pitch
(deg
)
0 2 4 6 8 10 12(s)
0
05
G
CG acc
minus050 2 4 6 8 10 12
(s)
(deg
)
0
100
200Yaw
0 2 4 6 8 10 12(s)
0
05
1
Met
er
Sway
0 2 4 6 8 10 12(s)
0
2
G
Bow acc
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
(c) Ten degrees roll angle
Figure 8 Effects of roll angle on planing craft motions for case 15
at similar roll angles Moreover due to the roll motion anintensive sway motion may occur Irregular heave and pitchmotions are also generated due to the asymmetric effect ofthe roll motion and the encounter wave These simulationsshow the reasonable behavior of the developed mathematicalmodel Itmust bementioned that variation of surge velocity isnot completelymodeled yet andwill be considered in the nextversion of the developed code Moreover some experimentalstudies should be conducted to further develop ormodify thepresented mathematical model
Nomenclature
119886 Wave amplitude119886119861119865 Buoyancy coefficient
119887 Instantaneous half beam of the section Time derivative of 119887119862119863 Drag coefficient
119862pu Splash-up coefficient119889119890 Effective depth of penetration
119889 Depth of penetration119891CD Viscous lift force associated with the cross
flow drag119891119872 Hydrodynamic lift force associated withthe change of fluid momentum per unitlength
119891119872PT 119891
119872associated with port side
119891119872SB 119891
119872associated with starboard side
119865119909 119865119910 119865119911 Force in 119909 119910 and 119911 directions
ℎ Submergence of a section119868119866119909119909
119868119866119910119910
119868119866119911119911
Moment of inertia in 119909119909 119910119910 and 119911119911
directions119896 Wave number119896119886 Added mass coefficient
119870119903 119875119903 119877119903 Hydrostatic moment in 119909119909 119910119910 and 119911119911
directions119870119908 119875119908 119877119908 Hydrodynamic moment in 119909119909 119910119910 and 119911119911
directions119870119891 119875119891 119877119891 Wave moment in 119909119909 119910119910 and 119911119911 directions
119897119908 Wetted length
119898 Mass119898119886 Added mass
119898119886PT
Added mass associated with port side119898119886SB Added mass associated with starboard side
119886 Time derivative of added mass
119872119909 119872119910 119872119911 Moment in 119909119909 119910119910 and 119911119911 directions
119905 Time119880 Velocity component parallel to the keel119881 Normal velocity Time derivative of normal velocity119908119911 Vertical component of the wave orbital
velocity at the surface
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 13
(deg
)
Roll
0
1
minus10 2 4 6 8 10 12
(s)
Met
er
Sway
0
1
minus10 2 4 6 8 10 12
(s)Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
(deg
)
Yaw
0
1
minus10 2 4 6 8 10 12
(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(a) Zero degree roll angle
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
5
(deg
)
Roll
minus50 2 4 6 8 10 12
(s)
0
02
04
Met
er
Sway
0 2 4 6 8 10 12(s)
0
50
(deg
)
Yaw
0 2 4 6 8 10 12(s)
CG acc
0
02
G
minus020 2 4 6 8 10 12
(s)
Pitch
4
6
8
(deg
)
0 2 4 6 8 10 12(s)
(b) Five degrees roll angle
Figure 9 Continued
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 International Journal of Engineering Mathematics
Heave
0 2 4 6 8 10 120
005
01
(s)
Met
er
Bow Acc
0
2
G
minus20 2 4 6 8 10 12
(s)
0
10
(deg
)
Roll
minus100 2 4 6 8 10 12
(s)
0
1
2
Met
er
Sway
0 2 4 6 8 10 12(s)
(deg
)
0
100
200 Yaw
0 2 4 6 8 10 12(s)
CG Acc
0
02
G
minus020 2 4 6 8 10 12
(s)
0
5
10 Pitch
(deg
)
0 2 4 6 8 10 12(s)
(c) Ten degrees roll angle
Figure 9 Effects of roll angle on planing craft motions for case 17
119911 Time derivative of 119908
119911
119909119862119866
119910119862119866
119911119862119866
Position of center of gravity (COG) of thehull in 119909 119910 and 119911 directions
119862119866
119910119862119866
119862119866
Velocity at COG in 119909 119910 and 119911 directions119862119866
119910119862119866
119862119866
Acceleration at COG in 119909 119910 and 119911
directions119883119903 119884119903 119885119903 Hydrostatic force in 119909 119910 and 119911 directions
119883119908 119884119908 119885119908 Hydrodynamic force in 119909 119910 and 119911
directions119883119891 119884119891 119885119891 Wave force in 119909 119910 and 119911 directions
(120577 120585) Coordinate system on the hull120588 Water density120591 Trim angle120573 Deadrise angle120578 Wave height120583 Angle between ship heading and wave
directionV Wave slope120596119890 Encounter frequency
120576 Phase angle120595 Roll angle Angular velocity of roll motion Acceleration of roll motion120579 Pitch angle120579 Angular velocity of pitch motion120579 Acceleration of pitch motion
120593 Yaw angle Angular velocity of yaw motion Acceleration of yaw motion
References
[1] Savitsky ldquoHydrodynamic design of planing hullrdquoMarine Tech-nology vol 1 no 1 pp 71ndash95 1964
[2] M Martin ldquoTheoretical prediction of motions of high-speedplaning boats in wavesrdquo Journal of Ship Research vol 22 no3 pp 140ndash169 1978
[3] E E Zarnick ldquoA non-linear mathemathical model of motionsof a planning boat in regular wavesrdquo Tech Rep DTNSRDC-78032 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1978
[4] E E Zarnick ldquoAnon-linearmathemathicalmodel ofmotions ofa planning boat in irregular wavesrdquo Tech Rep DTNSRDCSPD0867-01 David Taylor Naval Ship Reasearch and DevelopmentCenter Bethesda Md USA 1979
[5] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boatsrdquo Tech Rep 1275 Davidson Labora-tory Stevens Institue of Technology Hoboken NJ USA 1969
[6] G Fridsma ldquoA systematic study of the rough-water perfor-mance of planning boats(irregular wavesmdashpart II)rdquo Tech Rep11495 Davidson Laboratory Stevens Institue of TechnologyHoboken NJ USA 1971
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
International Journal of Engineering Mathematics 15
[7] J A KeuningThe nonlinear behaviour of fast monohulls in headwaves [PhD thesis] Technische Universiteit Delft Delft TheNetherlands 1994
[8] J D Hicks A W Troesch and C Jiang ldquoSimulation and non-linear dynamics analysis of planing hullsrdquo Journal of OffshoreMechanics and Arctic Engineering vol 117 no 1 pp 38ndash45 1995
[9] R H Akers ldquoDynamic analysis of planning hulls in the verticalplanerdquo in Proceedings of the Meeting of the New England Sectionof the Society of Naval Architects andMarine Engineers (SNAMErsquo99) Ship Motion Associates Portland Maine April 1999
[10] K Garme and A Rosen ldquoTime-domain simulations and full-scale trials on planing craft in wavesrdquo International ShipbuildingProgress vol 50 no 3 pp 177ndash208 2003
[11] K Grame and A Rosen Modeling of planning craft in waves[PhD thesis] Royal Institue of Technology KTH Departmentof Aeronautical and Vehicle Engineering Stockholm Sweden2004
[12] A van Deyzen ldquoA nonlinear mathematical model of motionsof a planning monohull in head seasrdquo in Proceedings of the 6thInternational Conference on High Performance Marine Vehicles(HIPER rsquo08) Naples Italy September 2008
[13] L Sebastianii D Bruzzone and P Gualeni ldquoA practical methodfor the prediction of planing craft motions in regular and irreg-ular wavesrdquo in Proceedings of the 27th International Conferenceon Offshore Mechanics and Arctic Engineering (OMAE rsquo08) pp687ndash696 Estoril Portugal June 2008
[14] H Sun and O M Faltinsen ldquoThe influence of gravity onthe performance of planing vessels in calm waterrdquo Journal ofEngineering Mathematics vol 58 no 1ndash4 pp 91ndash107 2007
[15] S B Rao andC K ShanthaNumericalMethodsWith Programsin Basics Fortran Pascal and C++ Universities Press IndiaRevised edition 2004
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of