477
1 INTRODUCTION
In the sea transportation of break bulk cargo,
particularly that of the non‐standard dimensions,
boththeweightofeachloadandtheactingforcesof
inertia should be considered when designing the
securingarrangementofthecargo.Theinertiaforces
can be evaluated once the linear accelerations
affectingthecargoareknown,whichdependon the
lawsoflineardisplacementchangesforthecargoand
the deck of the vessel relative to the reference
coordinatesystem.
Generally the oscillatory motion of the ship is
characterized by six degrees of freedom and is
described by six differential equations.
The ship
oscillationsarestronglycoupled[1].Itisshownin[2]
thatonecanapplythelinearmodelsofroll,pitchand
heave to obtain the linear acceleration in the first
approximation, i.e. use the corresponding isolated
equationsforthecalculation.
To be able to find the inertia forces acting
on a
cargoiteminaninertialreferenceframeitisenough
toapplytheNewtonʹs secondlaw,providedthatthe
massof the unitand therespective accelerationsare
known.Theaccelerationscanbefoundasthesecond
time derivatives of the angle of heel, the angle of
pitch,andtheamplitudeofheavefunctions.
Theexpressionsforthe angles and the amplitude
can be obtained by solving the equations for roll,
pitch,andheave.Thisisdoneontheassertionthatfor
thetaskoffindingtheforcesofinertiatheequations
forroll,pitch,andheavecan
beconsidereddecoupled
[3].
2 PARAMETERSOFOSCILLATIONS
2.1 Roll
As roll is the governing factor and generates
dominant forces of inertia further considerations
proceedwiththeequationthatdefinerollsolvingthe
problemsoastofindtheexpressionfortheangleof
heel
. For this, as suggested in [4], we use the
originalsecond‐orderlineardifferentialequationthat
definestherollangleofavessel
:
Determination of Inertia Forces Acting on Break Bulk
Cargo en Route
A.O.Chepok
OdessaNationalMaritimeAcademy,Odessa,Ukraine
ABSTRACT:Thepaperpresentstheanalyticalmethodofdefininginertiaforcesthatactonbreakbulkcargo
asaresultoftheoscillatorymotionofthevesselexposedtotheeffectofambientforces.Consideringthatthe
linearmodelsofroll,pitchandheave
applicableinthiscase,theproblemissolvedbyexpressingtheangleof
heel,theangleofpitch,andtheamplitudeofheave.Theobtainedfunctionsaredifferentiatedandtheinertia
forcesaredeterminedbymeansofapplyingtheNewtonʹssecondlaw.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 7
Number 4
December 2013
DOI:10.12716/1001.07.04.01
478
tDh=Dh++m+J
kxxx
sin
0θ0
(1)
whereJ
x=momentof inertiaof the vessel aboutthe
longitudinalaxisХ‐Х;m
x=generalizedaddedmasses
of water about the longitudinal axisХ‐Х;
x =
damping coefficient about the longitudinal axisХ‐Х;
D=displacementofthevessel(forceofgravity);h
0=
transverse initial meta‐centric height;
= reduction
coefficient for the roll oscillations;
k = the apparent
frequencyofthewaves.
Afterdividingtheequation(1)bythecoefficientof
thehighestderivativeweobtainthenormalisedform
oftheequation:
t=+h+
k
sin2
2
0θ
2
0
(2)
whererolldampingcoefficienth:
xx
x
m+J
=h
2
andeigenfrequencyoftherollingvessel
0:
xx
m+J
Dh
=
0
2
0
The expression (2) is a linear non‐homogeneous
differentialequationwithconstantcoefficients,andits
solutionisthesumofaparticularsolution
r,which
describestheforcedoscillationofthevesselaboutthe
axis X‐X influenced by the regular waves, and the
solutions of the corresponding homogeneous
equation,whichdescribesowndampedoscillationsof
theship.
Since the amplitude of the vesselʹs own damped
oscillationsturnstozeroratherquickly,the
equation
ofroll,asastationaryprocess,accordingto[4]canbe
describedasforcedoscillationsonly,i.e.:
22
0
0,5
22
2
22
0
2
0θ
2
sin
4
k
k
k
kk
h
arctgt
h+
=
(3)
2.2 Pitch
Similarlytothecaseofroll,asitwasdemonstratedin
theworks[3,4]shipperformsforcedoscillationswith
the frequency of
k while pitching. The isolated
equation of longitudinal pitching, as well as its
solution,hasstructure similar tothestructure ofthe
transverserollingequation,i.e.describesnotonlythe
vesselʹsowndampedoscillations,butalsotheforced
harmonic oscillations with the pitch frequency. This
waytheexpressionthatdefining
thecurrentangleof
trim
is similarly characterized by the induced
harmoniousvibrationswiththepitchfrequency
k:
22
β0
β
0,5
22
β
2
22
β0
2
β0β
2
sin
4
k
k
k
kk
h
arctgt
h+
=
(4)
where
= reduction coefficient for the roll
oscillations;
k = eigenfrequency of the pitching
vessel;h
=pitchdampingcoefficient.
2.3 Heave
Finally,heaveistheresultoftheorbitalmotionofthe
vesselonaradiusequaltothehalfofthewaveheight
[3, 5]. Heave motion
has harmonic character with
thefrequencyofoscillations
kandcanbedescribed
asfollows:
t=
k
sin
0
(5)
where
0=amplitudeofthevertical motion induced
bythewaveswiththeheightofh
w:
w
h= 0,5
0
(6)
3 FORMULATINGTHEINERTIAFORCES
Theresultingexpressions (3),(4)and(5)allowus to
calculate the angular accelerations of the roll and
pitch,thelinearaccelerationandinertiaforcesacting
on the cargo. From this we find the inertia forces
inducedby roll,pitch andheave that act
ona cargo
unitwiththemassm
c.
Themost substantial is the lateralforce ofinertia
oftherollF
.Itisobviousthat:
yc
am=F
θ
wherea
y=linearaccelerationduetoroll.
Initsturn,thelinearaccelerationa
yistheproduct
of the angular acceleration
by the radius of
curvature r
y relative to the longitudinal axis passing
throughthecenterofgravityofthevesselG,i.e.:
yy
r=a
Thus finding the angular acceleration
as the
secondderivative of the roll angle by differentiating
twicetheexpression(3)yields:
t=
kk
sin
2
0
where
479
0,5
22
2
22
0
2
0θ
0
4
kk
h+
=
22
0
2
k
k
h
arctg=
SuccessivelytheinertiaforceF
isdefinedas:
trm=F
kkyc
sin
2
0θ
(7)
ThelongitudinalforceF
canbederivedinmuch
the same way, i.e. F
r=‐mcax where ax = linear
accelerationduetopitch.
The linear acceleration in this case is
xx
r=a
where r
x = radius of curvature relative to the
transverse axis. The angular acceleration
can be
obtainedbydifferentiatingtheexpression(4)twice:
β
2
0
sin
t=
kk
0,5
22
β
2
22
β0
2
β0β
0
4
kk
h+
=
22
β0
β
β
2
k
k
h
arctg=
ThereforetheforceofinertiaF
isrepresentedby
theequation:
β
2
0β
sin
trm=F
kkxc
(8)
TheheavingforceofinertiaF
isformulatedas
F
=‐mc
Linear acceleration
we get as the second
derivativeoftheexpression(5):
t=
kk
sin
2
0
Then, taking into account equation (6) we finally
putF
as:
tmh=F
kkcw
sin0,5
2
ζ
(9)
ItistobenotedthattheinertiaforcesF
,F
,andF
wereobtainedwiththereferencetotheunperturbed
system of coordinates. Then in order to be able to
calculate the reactions in lashings of the cargo these
inertial forces and the force of gravity P
c must be
projected on to the shipʹs frame of axes which is
inclinedbytheanglesofheel
andtrim
asshown
intheFigure1.
Figure1. Frame of axes referenced to the unperturbed
systemofcoordinates.
4 REMARKSANDCONCLUSIONS
The article describes the method of deriving the
inertiaforcesactingonacargounitsotheseforcescan
beaccountedforinfurthercalculationstodetermine
themaximumworkingloadoflashingsforthecargo.
The method is based on the presumption that the
linearmodels
ofroll,pitchandheavearesufficientfor
the case and can be considered independent within
thescopeoftheproblem.Theresultingexpressionsof
theangleofheel,theangleoftrimandtheamplitude
ofverticalmotioninducedbywavesallowcalculating
the respective angular and linear accelerations.
The
inertia forces determined in the unperturbed
reference frame can be easily ported to the shipʹs
systemofcoordinatesastherelationbetweenthetwo
systems of coordinates is known. The obtained
functionsareusedbytheauthorinhismathematical
model describing the process of safe stowage and
lashing
ofbreakbulkcargoonboardaship.
REFERENCES
1.Kornev N. 2011. Ship dynamics in waves. – Rostock:
UniversityofRostock.
2.Borodaj I.K., Necvetaev Ju.A. 1969. Kachka sudov na
morskomvolnenii.–L.:Sudostroenie.
3.SizovV.G.2003.Teorijakorablja.–Odessa:Feniks.
4.Vojtkunskij I.J. 1985. Statika sudov. Kachka sudov. In
Spravochnik po teorii korablja: V 3
‐h t. – L.:
Sudostroenie.
5.Shimanskij J.A. 1948. Dinamicheskij raschet sudovyh
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