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공학석사 학위논문
Motion Analysis of the Riser on
the Drilling Rig using Flexible
Mutibody Dynamics
유연 다물체계 동역학을 이용한 해양 시추 리그의
라이저 거동 해석
2016 년 2 월
서울대학교 대학원
조선해양공학과
홍 정 우
I
Abstract
Motion Analysis of the Riser on the Drilling
Rig using Flexible Multibody Dyanmics
Dynamic analysis is broadly used in offshore and ship building industry for
determining dynamic loads of a block lifting and a subsea equipment
installation operation, and a coupled analysis of the offshore drilling rig and a
riser. If the bodies in simulated system regard as rigid bodies, the results of the
simulation result cannot reflect the behavior due to the deformation of the
bodies. This leads that we cannot simulate the system that has large deformation
bodies which affect to the behavior of the system such as coupled analysis of a
platform and a riser. Therefore, deformation of the bodies have to be considered
in dynamic analysis. From this reason, this study investigated Absolute Nodal
Coordinate Formulation which is one way of the flexible multibody dynamics
that takes account of the deformation of the bodies in the multibody system.
The developed flexible multibody dynamics code is verified with analytic
soluation and the commercial code, and applied to the motion analysis of a
drilling rig and a riser.
For the modeling of a drilling rig with a riser, a semi-submersible platform,
telescopic joints that connects the platform and ther riser, a lower flex joint that
connects the riser and the BOP(Blow Out Preventer) are modeled as rigid body,
and the riser is modeled as ANCF(Absolute Nodal Coordinate Formulation)
beam. The wire-line riser tensioner which pull the riser to maintain tension of
riser to prevent buckling, and keep the vertical position of the riser is modeled
as pneumatic spring. Each body is interconnected by some types of joints such
II
as fixed joint, cylindrical joint, and ball joint. The hydrostatic and
hydrodynamic force on the platform, current force on the riser, gravity on the
all bodies are considered as external forces. Finally, the coupled motion
analysis of a drilling rig and a riser has performed by using flexible multibody
dynamics.
Keywords: Flexible multibody dynamics, Multibody dynamics, Riser,
Offshore drilling rig, Dyanmic behavior, Absolute Nodal Coordinate
Formulation.
Student number: 2014-20650
III
Tables of contents Abstract .......................................................................................... I
Tables of contents ....................................................................... III
List of tables ................................................................................ VI
List of figures ............................................................................. VII
Introduction ............................................................................. 1
1.1. Background .............................................................................................. 1
1.2. Research objective ................................................................................ 2
1.3. Related works .......................................................................................... 4
1.4. Contributions of this study ................................................................ 5
Equations of the motion of the flexible multibody system 6
2.1. Concept of flexible multibody dynamics ..................................... 6
2.1.1. Floating Frame of Referecne Frame formulation (FFRF) . 7
2.1.2. Kinematic description of deofrmable body using Absolute Nodal Coordinate Formulation (ANCF) ......................... 9
2.2. Equations of the motion of a beam element using ANCF ..11
2.2.1. Kinematic description of a beam element .........................11
2.2.2. Mass matix of a beam element ..............................................12
2.2.3. Stiffness matrix of a beam element ......................................14
2.3. Equations of the motion of a plate element using ANCF ...25
2.3.1. Kinematic description of a plate element ...........................25
2.3.2. Mass matrix of a plate element ..............................................30
2.3.3. Stiffness matrix of a plate element ........................................32
2.4. Derivation of the general form of the equations of the motion of the flexible multibody system ...........................................41
2.5. Kinematic constraints .........................................................................44
IV
2.5.1. Rigid body – Rigid body constraints ....................................44
2.5.2. ANCF beam – ANCF beam constraints ................................52
2.5.3. ANCF beam – Rigid body constraints ..................................54
2.6. External forces .......................................................................................59
2.6.1. Hydrostatic force ..........................................................................59
2.6.2. Hydrodynamic force ....................................................................59
2.6.3. Gravity on the ANCF body ........................................................61
2.6.4. Current force on the ANCF body ...........................................62
2.7. Method of numerical integration ..................................................63
2.8. Configurations of the simulation program ...............................65
Motion analysis of a drilling rig with a riser ...................... 67
3.1. Verification of flexible elements ....................................................67
3.1.1. Verification of a beam element ...............................................67
3.1.2. Verification of a plate element ................................................69
3.2. Verification of a simulation program with OrcaFlexTM ..........72
3.2.1. Modeling of the verification example ..................................72
3.2.2. Simulation results of the verification example .................74
3.3. Modeling of the drilling rig with a riser ......................................88
3.3.1. Configurations of the drilling rig with a riser ....................88
3.3.2. Wireline riser tensioner ..............................................................89
3.3.3. Modeling information of the drilling rig with a riser .....92
3.4. Numerical simulation results of a drilling rig with a riser....95
3.4.1. 10 elements without current....................................................95
3.4.2. 10 element with 3 m/s current ............................................. 102
Conclusions and future works ............................................ 109
4.1. Conclusions ......................................................................................... 109
4.2. Future works ....................................................................................... 109
References ................................................................................ 111
V
Appendix .................................................................................. 113
Derivation of the curvature in small longitudinal deformation ................................................................................................ 113
Explicit matrix expression for plate element .................................. 115
Basic notation ......................................................................................... 115
Expressions for Mid-Plane Forces ................................................... 115
Expressions for Transverse Forces .................................................. 117
국문 초록 ............................................................. 119
VI
List of tables Table 1 Related works of the coupled analysis of a platform and a riser ......... 4
Table 2 Comparison of FFRF and ANCF ....................................................... 10
Table 3 Specifications of a 2D cantilever beam ............................................. 68
Table 4 Specifications of a plate element ....................................................... 69
Table 5 Properties of the barge used in the verification example ................... 73
Table 6 Properties of the riser used in the verification example .................... 73
Table 7 Body information of the main example ............................................. 92
Table 8 Joint information of the main example .............................................. 93
Table 9 Properties of the semi-submersible platform used in the main example .................................................................................................................... 94
Table 10 Properties of the riser used in the main example ............................. 94
VII
List of figures
Fig. 1 Examples of dynamic analysis required situations ................................ 1
Fig. 2 Four coordinate systems used in floating frames of reference formulation ................................................................................................... 7
Fig. 3 Summary of the kinematic description of the FFRF .............................. 8
Fig. 4 Arbitrary position vector on the deformable body using ANCF ............ 9
Fig. 5 Kinematic description of an ANCF beam element ............................... 11
Fig. 6 Definition of the angles between the beam and the axix ..................... 18
Fig. 7 An arbitrary position vector on the plate element surface .................... 25
Fig. 8 Nodal coordinate vectors of a ANCF plate element ............................ 26
Fig. 9 Nodal coordinates to determine 1 on 2p parameter beam. .......... 27
Fig. 10 Nodal coordinates to determine 1 on 2p parameter beam. .......... 28
Fig. 11 Nodal coordinates to determine 2 on 2p parameter beam. ........ 29
Fig. 12 Nodal coordinates to determine 2 on 2p parameter beam. ......... 29
Fig. 13 An arbitrary position vector on the rigid body ................................... 45
Fig. 14 Definition of Euler angles .................................................................. 46
Fig. 15 Kinematics of a ball joint ................................................................... 47
Fig. 16 Kinematics of a cylindrical joint ........................................................ 49
Fig. 17 Kinematics of a fixed joint ................................................................. 51
Fig. 18 Kinematic description of an ANCF beam element (Fig. 5) ............... 52
Fig. 19 Kinematics of an ANCF beam and a rigid body ................................ 54
Fig. 20 Kinematics of a fixed joint between an ANCF beam and a rigid body .................................................................................................................... 56
Fig. 21 Calculation procedure of hydrodynamic forces ................................. 60
Fig. 22 Configurations of the simulation program ......................................... 65
Fig. 23 Simulation sequence of the anlaysis program .................................... 66
Fig. 24 Deformation of the 2D cantilever beam ............................................. 68
Fig. 25 Convergence test of the numerical cantilever beam model................ 68
Fig. 26 Dimension of a plate element ............................................................ 70
Fig. 27 Convergence test of the pure bending of numerical plate model ....... 70
VIII
Fig. 28 Configurations of the modeling of the verification example ............. 72
Fig. 29 Screen shot of the simulation by in-house code and OrcaFlexTM ...... 74
Fig. 30 Comparison of the heave motion of the platform with 1 element riser .................................................................................................................... 74
Fig. 31 Comparison of the heave motion of the platform with 5 element riser .................................................................................................................... 75
Fig. 32 Comparison of the behavior of one element riser .............................. 81
Fig. 33 Comparison of the behavior of five elements riser ............................ 86
Fig. 34 Configurations of the drilling rig with a riser .................................... 88
Fig. 35 Model of pressure system for riser tensioner ..................................... 89
Fig. 36 Configurations of a wireline riser tensioner and a riser ..................... 90
Fig. 37 Bodies and joints information of a drilling rig with a riser ................ 92
Fig. 38 Screen shot of the drilling rig simulation ........................................... 95
Fig. 39 Surge motion of the platform without current.................................... 95
Fig. 40 Heave motion of the platform without current ................................... 96
Fig. 41 Roll motion of the platform without current ...................................... 96
Fig. 42 Behavior of the riser without current ............................................... 101
Fig. 43 Surge motion of the platform with 3 m/s current ............................. 102
Fig. 44 Heave motion of the platform with 3 m/s current ............................ 102
Fig. 45 Roll motion of the platform with 3 m/s current ............................... 102
Fig. 46 Behavior of the riser with 3m/s current ........................................... 107
1
Introduction 1.1. Background
Fig. 1 Examples of dynamic analysis required situations
Dynamic analysis is broadly used in offshore and ship building industry for
determining dynamic loads of a block lifting and a subsea equipment
installation operation, and a coupled analysis of the offshore drilling rig and a
riser as Fig. 1. If the bodies in simulated system regard as rigid bodies, the
results of the simulation cannot reflect the behavior due to the deformation of
the bodies. This leads that we cannot simulate the system that has large
deformation bodies which affect to the behavior of the system such as coupled
analysis of a platform and a riser. Therefore, deformation of the bodies have to
be considered in dynamic analysis. From this reason, this study investigated the
flexible multibody dynamics that takes account of the deformation of the bodies
in the multibody system and applied to the motion analysis of a drilling rig and
a riser.
2
The requirements of this study are listed as bellows.
(1) Convinience of formulation of the equations of the motion
Systems that we analize are usually consist of interconnected bodies. To
build equations of motion of the interconnect bodies effectively, multibody
dynamics is used to get the behaviors and the constraint forces in the system.
However, this work requires many efforts. Therefore, if the equations of the
motion of the dynamic system is formulated automatically from the information
of the bodies and joints, we can save the time and effort to build equations of
the motion.
(2) More realistic simulation results.
In reality, bodies deform due to the huge load on the body or dynamic load
when the body is highly elastic. This deformation affects to the behavior of the
entire system. Therefore, we can predict the deformation of the body and the
behavior of the system by using flexible multibody dynamics.
(3) Dynamic structural analysis
The basic concept of the flexible mutibody dynamics is derived from the
finite element method. Thus, flexible multibody dynamics can predict the
maximum stress while in the motion, because the flexible multibody dynamics
can get the change of the stresses that take account of both the static load and
the dynamic load in real time. The flexible mutlbody dynamics shows the
vibrations of the system which include the natural periods of the bodies, and it
makes one can avoid the resonance of the system.
1.2. Research objective
3
Research objectives of this study are listed as bellows
(1) Derivation of the equations of the motion of a beam element by using
flexilble multibody dynamics.
(2) Derivation of the equations of the motion of a plate element by using
flexilble multibody dynamics.
(3) Derivation of the joint constraints used in flexible multibody dynamics
(4) Automation of (1), (2) and (3)
(5) Applying (1) ~ (4) to the motion analysis of a drilling rig with a riser.
4
1.3. Related works Table 1 Related works of the coupled analysis of a platform and a riser
Multibody
formulationCoupled analysis
Platform type
Riser model
Garret (2005)
X O Semi-
submersibleFE beam
Low and Langely (2008)
X O Ship Spring
Santillan et al. (2010)
X X N/A Numerical
beam
Yang et al. (2012)
X O Spar FE beam
Park (2011) O N/A Wind
turbine N/A
OrcaFlexTM △ O N/A FE beam (lumped mass)
This study O O Semi-
submersible
FE beam (consistent
mass)
Table 1 shows the related works of the coupled analysis of a platform and a
riser. Garret (2005) suggested coupled motion of the riser and semi-submersible
with finite element beam riser model with boundary conditions. Low and
Langely (2008) solved coupled motion of the riser and a ship with spring riser
model. Santillan et al. (2010) suggested differential equations of the steel
cartenary riser that derived from its geometry. Yang et al. (2012) solved
coupled motion of the riser and spar with finite element beam riser model and
compared with experimental result. Park (2011) introduced flexible multibody
dynamics to the shipbuilding and offshore industry with a motion analysis of a
wind turbine. OrcaFlexTM is a commercial code that supports coupled analysis
with lumped mass finite element model. This study uses flexible multibody
dynamics for solving coulpled analysis of a drilling rig and a riser.
5
1.4. Contributions of this study
Contributions of this study is listed as belows
(1) This study analyze the coupled motion of the dillring rig with a riser, and
it helps prediction of the maximum deformation and the stress of the riser
in various environmental condition.
(2) This study can suggest design criteria of riser using the result of (1).
(3) The time domain analysis of a riser can be used for the fatigue analysis.
(4) The developed flexible multibody dynamics code can be used in various
engineering fields that required dynamic analysis of the deformable
bodies such as beam and truss analysis of offshore structures, and
mooring line analysis.
6
Equations of the motion of the
flexible multibody system
2.1. Concept of flexible multibody dynamics
Multibody dynamics is the discipline describing the dynamic behavior of
mechanical systems which consist of several bodies connecting with kinematic
constraints called joints that impose restrictions on their relative motion.
Multibody systems can be divided into three cartegories, rigid multibody
systems, linearly elastic multibody systems, and nonlinearly elastic multibody
systems. (Bauchaue, 2011) Rigid multibody systems consis of an assemblage
of rigid bodies connected together through mechanical joints and in arbitrary
motion with respect to each other. Linearly elastic multibody systems sonsist
of an assemblage of both elastic and rigid bodies connected together through
mechanical joints and in arbitrary motion with respect to each other. For
linearly elastic multibody systems, it is assumed that the strain-displacement
relation shipe remain linear and that straints componetents remain very small
at all times for elastic bodies. Efficient analysis techniques for this type of
problems is Floating Frame of Refrence Frame formulation (FFRF).
Nonlinearly elastic multibody systems consist of an aseemblage of both elastic
and rigid bodies connected together through mechacnical joints and in arbitrary
motion with respect to each other. For the elastic bodies, the strain-
displacement relationship become nonlinear, or the strain components become
large, or both. Efficient analysis tecciques for this type of problem is Absolute
Nodal Coordinate Formulation (ANCF) which is mainly handled in this study.
Following chapters explane about two different formulation FFRF and ANCF.
7
2.1.1. Floating Frame of Referecne Frame formulation
(FFRF)
Fig. 2 Four coordinate systems used in floating frames of reference formulation
Floating Frame of Reference Frame formulation (FFRF) is the one of the
technique for linear elastic multibody system. FFRF uses four coordinates
systems to describe arbitrary points on a deformable body in the global
coordinate system. Fig. 2 shows the four coordinates systems used for the
formulation. The superscript i refers to the body number in the multibody
system; the superscript j , to the element number in the finite element
discretization of the deformable body i ; and the subscript i , to the intermediate
element coordinate system. A global coordinate system is fixed in time and
forms a single standard for the entire assembly of bodies, and thus, expresses
the connectivity of all the bodies through the system. A body coordinate system
forms a single standard for the entire assembly of elements in the body i and
thus expresses the connectivity of all the elements in the body. An intermediate
element coordinate system is a system whose origin is rigidly attached to the
origin of the body coordinate system and does not follow the deformation of
the element. The intermediate element coordinate system is initially oriented to
8
be parallel to the element coordinate system. An element coordinate system is
rigidly attached to each element j on the deformable body i . This coordinate
system translates and rotates with the element.
In FFRF, an arbitrary position vector on a deformable body is defined as
equation (1)
1
1 2( )
ij i i ij
i i ij ij ij ij in
i i ij ij ij ij i i io f
r R A u
R A C S C B q
R A C S C B q B q
(1)
wherein iR is the position vector of the orientation of the body coordinate
system with respect to the global coordinate system, iA is the rotation matrix
between the body coordinate system and the global coordinate system. ijS is
the element shape matrix, ijC and ijC are rotation transformation matrces
between body coordinates system and intermediate element coordinates system,
1ijB is the element selection matrix, and finally i
nq be the total vector of the
nodal coordinates of body i which consist of vector of the nodal coordinates
in the undeformed state ioq and the vector of the nodal deformation i
fq
(Shabana, 2005).
Fig. 3 Summary of the kinematic description of the FFRF
9
Fig. 3 shows summary of the kinematic description of the flexible body with
their definition. This kinematic description method uses 4 different coordinate
systems from the nodal coordinate system with respect to the body coordinate
system to the global coordinate system via intermediate element coordinate
system and element coordinate system.
2.1.2. Kinematic description of deofrmable body using
Absolute Nodal Coordinate Formulation (ANCF)
Fig. 4 Arbitrary position vector on the deformable body using ANCF
Absolute Nodal Coordinate Formulation (ANCF) is the one of the technique
for nonlinear elastic multibody system. ANCF can be used in the large rotation
and deformation analysis of flexible bodies that undergo arbitrary displacement.
In this formulation, no infinitesimal or finite rotations are used as the nodal
coordinates. Instead, absolute solopes and displacements at the nodal points are
used as the element nodal coordinates (Shabana, 2005).
In ANCF, the nodal coordinates and slope are defined in the inertial frame.
These nodal coordinates are used with a global shape function that has a
complete set of rigid body modes. Therefore, the global position vector of an
10
arbitrary point on the element can be described using the global shape function
and the absolute nodal coordinates as Fig. 4 and equation (2)
er S , (2)
wherein S is the global shape function and e is the vector of element nodal
coordinates.
Table 2 Comparison of FFRF and ANCF
FFRF ANCF
Coordinate system 4 reference frames
coordinates Absolute coordinates
Inertia matrix Non-linear Constant
Stiffness matrix Constant Non-linear
Quadratic velocity term
O X
Large deformation X O
Table 2 is the comparison of FFRF and ANCF. FFRF is derived based on
the body and element coordinates system including large rotation vector
formulations leads to a highly nonlinear mass matrix and quadratic velocity
terms which include Corioilis force. However, deformation vectors are
represented in the nodal coordinates this leads to a stiffness matrix is constant.
These deformation vectors are linearlized and thus FFRF is inappropriate for
large deformation problem. ANCF is derived based on the global coordinates
system with displacement and slope of the nodes. This lead to mass matrix of
ANCF constant, and there is no quadratic velocity term. However, description
of deformation is not intuitive in globar coordinates system, and this makes
stiffness matrix of ANCF is nonlinear.
11
2.2. Equations of the motion of a beam element
using ANCF
This chapter derives equations of the motion of a beam element using ANCF.
2.2.1. Kinematic description of a beam element
Fig. 5 Kinematic description of an ANCF beam element
To describe beam element with ANCF, we need information about
displacement and slope of nodes like Fig. 5. Therefore, a nodal coordinates of
a ANCF beam e is consist of displacement vector 0r and lr which are the
displacement of the nodes, and 0r and lr which are the slopes of the nodes.
Using this features, equation (2), can be re-written as
0
01 2 3 4
l
l
p s s s s
r
rr I I I I
r
r
, (3)
wherein I is 3 by 3 identity matrix and is ( 1,2,3,4i ) are the Hermite
shape functions defined as
12
2 31 3, , 1 3 2s p l s l p l ,
2 32 4, , 2s p l s l p l l ,
2 33 , 3 2s p l ,
3 24 ,s p l l
where, .p
l
(4)
l is the length of the beam, and p is the parameter of the archlength of the
beam. Hermite shapefunction interpolates the beam element as a cubic
polynomial. Explicit form of the shape function S can be written as
2 3 2 3
2 3 2 3
2 3 2 3
2 3 3 2
2 3 3 2
2 3 3 2
1 3 2 0 0 2 0 0
0 1 3 2 0 0 2 0
0 0 1 3 2 0 0 2
3 2 0 0 0 0
0 3 2 0 0 0 .
0 0 3 2 0 0
l
l
l
l
l
l
S
(5)
2.2.2. Mass matix of a beam element
Mass matrix have to be defined to drive equations of the motion of the
flexible body. Mass matrix is defined from a kinetic energy of ANCF body
which is defined as
T1
d2 V
T V r r (6)
wherein is a density of a beam, and r is a velocity vector which is time
derivative of equation (2). Velocity vector r is define as
13
d d d
d d dt t t
r Se er S Se (7)
From the definition of the kinetic energy, equation (6) can be re-written as
T T1 1
d2 2V
T V r r e Me . (8)
From the equation (8), a mass matrix M is defined as
T d
V
V M S S . (9)
Explicit form of the mass matrix of ANCF beam is as
2 2
2 2
2 2
2
2
13 11 9 130 0 0 0 0 0 0 0
35 210 70 42013 11 9 13
0 0 0 0 0 0 035 210 70 420
13 11 9 130 0 0 0 0 0
35 210 70 420
130 0 0 0 0 0
105 420 140
130 0 0 0 0
105 420 140
130 0 0 0
105 420 14013 11
0 0 0 035 210
13 110 0 0
35 21013 11
0 035 210
0 0105
l l
l l
l l
l l l
l l l
l l l
l
l
l
l
l
M
2
0105
105
l
.
(10)
With result of equation (10), we verify that the mass matrix of an ANCF
body is constant matrix.
14
2.2.3. Stiffness matrix of a beam element
Stiffness matrix of a beam element is obtained from potential energy due to
deformations. Potential energy of a beam is consist of 3 components with 3
dimensional beam element which are longitudinal, bending, torsional strain
energy. This study neglects torsional effect of a beam, uses deformation beam
model developed by Berzeri et al. (2000).
The strain energy lU due to the longitudinal deformation is defined as
2
0
1= d
2
l
l lEA xU ε (11)
wherein E is Young’s modulus, and l is Green-Lagrange strain tensor
which is defined as
1 11 1
2 2
TT
l
r rε r r
x x. (12)
The strain energy tU due to the bending is defined as
2
0
1= d
2
l
t EI xU κ (13)
wherein I is the second moment of the area, and κ is the curvature of the
beam which is defined as
3
r rκ
r. (14)
Therefore, total strain energy U due to the deformation of the beam is sum
of lU and tU as
2 2
0
1 = + = d
2
l
l t lEA EI x U U ε U κ . (15)
15
(1) Longitudinal force models
Several longitudinal force models which can be used in ANCF are presented.
Although these models account for the elastic non-linearities as described by
the Green-Lagrage strain tensor, two of them lead to expressions simpler than
the one obtained by the model developed using the local element coordinate
system and a linear strain-displacement relationship.
From equation (12), following matrices lS and lS are defined for
convenience
T T
2
1l l S S S S S ,
1
0
dl l S S (16)
wherein S is the derivative of the shape function S with respect to the
parameter x
l . Using equation (2), equation (17) is obtained.
T T T r r e S S e , (17)
Substituting equation (17) in to equation (12), and using the equation (16),
the longitudinal strain can be written as
T11
2l l ε e S e (18)
The derivative longitudinal strain with respect to the nodal coordinates is
obtained as
T
ll
εS e
e (19)
Using equation (19), generalized elastic force due to the longitudinal
deformation lQ is defined as
16
T
0
dl
ll l l
UEA x Q S e
e, (20)
And also lQ can be written as equation (21) which defines the stiffness
matrix due to the longitudinal deformation lK as equation (22).
l lQ K e (21)
0
dl
l l lEA x K S (22)
Depending on the way the strain l , differenent longitudinal force models
can be derived as L1 and L2 model.
① L1 model
In case of small deformation, the strain l is assumed to be constant
thoughout the beam element, then possible to l out of the integral sign of
equation (22). Using this asuumption and assuming that E and A are
constant, the stiffness matrix due to the longitudinal lK can be written as
0
dl
l l l l lEA x EA K S S . (23)
In equation (23), l is the average longitudinal strain along the element,
and in the case of small deformation l is simply approximated as
l
d l
l (24)
wherein d is the distance between the nodes of the element defined as
17
2 2 2
7 1 8 2 9 3d e e e e e e (25)
wherein ie are the component of the nodal coordinate vector. The nodal
coordinate vector e can be written as equation (26) for the beam element in
the 3 Dimensional space.
1 2 3 4 5 6 7 8 9 10 11 12e e e e e e e e e e e ee (26)
Using the constant strain assumption as equation (24), the stiffness matrix
lK can be writeen explicitly as
2 2
2 2
2 2
2
2
2
6 60 0 0 0 0 0 0 0
5 10 5 106 6
0 0 0 0 0 0 05 10 5 10
6 60 0 0 0 0 0
5 10 5 10
20 0 0 0 0 0
15 10 30
20 0 0 0 0
15 10 30
20 0 0 0
15 10 306
0 0 0 05 10
60 0 0
5 106
0 05 10
20 0
15
20
15
2
15
l l
l l
l l
l l
l l l
l l l
l l lEA
ll
l
l
l
l
l
K
(27)
18
② L2 model
L2 model is general expression for the longitudinal strain that has no
assumption. For L2 model, the nodal coordinate vector e is written as the sum
of two vectors as
r f e e e (28)
wherein re represent an arbitrary rigid body displace ment. In case of a beam
element in 3 dimensional space, re is defined as
sin cos sin sin cos
sin cos s s cos sin cos sin sin cos
r x y z
x l y l z l
e (29)
wherein the angles between the beam and the axis and are defined as
Fig. 6.
Fig. 6 Definition of the angles between the beam and the axix
19
Using equation (28), one can find a identity that as,
T 1r l r e S e (30)
the quantity of Tle S e becomes very close to one when the deformation is very
small. However, Substituting equation (30) into right hand side of equation (18),
the longitudinal strain becomes as,
TT T1 1
2 2l l r l r r l r ε e S e e S e e e S e e (31)
which shows that with an arbitrary convinient choice for re , the strain l
determined accurately. Using equation (22) and assuming E and A are
constant, equation (22) can be writtens as
0
dl
l l lEA x K S . (32)
Since choose 0 0 0 1 0 0 0 0 1 0 0r le for
convinient calculations, and substitute re and equation (26) into equation (31),
the stiffness matrix of equation (32) can be calculated as
A Β -A C
D -B E
A C
. F
l
EA
l
sym
I I I I
I I IK
I I
I
(33)
wherein I is 3 by 3 identity matrix, and six independent elements A , B , C ,
D , E , and F are expressed with these quantities in equation (34),
2 2 27 1 4 5 6
2 2 28 2 10 11 12
2 2 29 3
, , , ,
, , , ,
,
x x y z x y z
y x y z x y z
z x y z
d e e a le a le a le a a a a
d e e b le b le b le b b b b
d e e d d d d
(34)
and A , B , C , D , E , and F are as
20
2 2 2 22
2 2 2 2
2 2 2 2
2 2
3A 14 6 6 6 6 6 6 24
701
B 2 2 2 14 24 24 24 36280
1C 2 2 2 14 24 24 24 36
2801
D 12 3420
x x x x y y y y z z z z
x x y y z z x x y y z z
x x y y z z x x y y z z
x x
a b l a d b d a d b d a d b d dl
b a a b a b a b l a d a d a d dl
a b a b a b a b l b d b d b d dl
a b a b
2 2
2 2 2
2 2 2
3 3 28 3 3 3 3 3 3 18
1E 3 3 4 4 4 14 6 6 6 6 6 6
8401
F 12 3 3 3 28 3 3 3 3 3 3420
y y z z x x y y z z x x y y z z
x x y y z z x x y y z z x x y y z z
x x y y z z x x y y z z x x y y z
a b a b l a d a d a d b d b d b d d
a b a b a b a b l a d a d a d b d b d b d
a b a b a b a b l a d a d a d b d b d b
218zd d
.
(35)
Equation (35) shows that the stiffness matrix of the ANCF beam is a full
matrix whose elements are quadratic functions of the nodal coordinate ie .
(2) Tranverse force models
There are two models for the elastic forces associated with the transeverse
deformation are developed for the ANCF. The first model T1 can be used when
the longitudinal deformation is small. The second model T2 can be used in case
of large longitudinal and transe verse motion.
① T1 model
For T1 model, the curvature in equation (14) can be simplified as equation
(36) in case of small longitudinal deformation.
2
3 2
d
dx
r r rκ r
r (36)
Details about derivation of equation (36) is in Appendix. Using equation (2),
equation (36) can be written as
2
2x
rκ S e , (37)
and square of the curvature is obtained as
21
222
2T T
x
rκ e S S e . (38)
The strain energy defined by equation (13) in T1 model can be written as
2
0 0
1 1 = d d
2 2
l l T Tt EI x EI x U κ e S S e . (39)
From Euqtion (39), the stiffness matrix due to the transverse deformation
tK can be defined as foloows
T1 =
2t tU e K e , (40)
0
dl T
t EI x K S S . (41)
Assuming E and I are constant and using the shape function of
equation (5), the stiffness matrix tK can be written explicitly as
2 2
3
2
12 6 12 6
4 6 2
12 6
. 4
t
l l
l l lEI
ll
sym l
I I I I
I I IK
I I
I
, (42)
wherein I is 3 by 3 identity matrix.
② T2 model
When the longitudinal deformation is large, the used of equation (36) cannot
follow the exact curvature of the beam, as it is clear from equation (14), that
shows the curvature can significantly change in the case of a large longitudinal
deformation. From the equation (15), curvature of the ANCF bodies can be
written as matrix form as
22
3/2 3T
1TT
f
r Irκ r Ir
r r
, (43)
wherein f defined as equation (44) and the I defined as equation (45)
which converts vector cross product into matrix form.
Tf r r (44)
0 1 1
1 0 1
1 1 0
I (45)
In equation (42), the term T r Ir can be calculated as
T ˆTt r Ir e S e , (46)
wherein ˆtS is defined as
T3
1ˆ Tt l S S IS S IS . (47)
The curvature is scalar, thus it is equal to its transpose, therefore equation
(48) is valid
T Tˆt te S e e S e (48)
wherein the symmetric matrix tS is defined as,
T1 ˆ ˆ2t t tt S S S (49)
Using Eqution (48), the curvation can be written as
T
3t
f
e S eκ . (50)
This equation can be substituted into the strain energy tU of equation (13)
to get the elastic forces due to the transverse deformation. However, equation
(50) leads to a complex expression for the elastic forces. If the longitudinal
deformation within the element is assumed constant when derive transverse
elastic force, this can be simplified. To do this, the deformation gradient f is
23
assumed to an equivalent constant value f as
d
.d
sf const
x (51)
It follows
2
2
d 1
ds f
rr , (52)
and the square of the curvature becomes as
2
4
1 T T
f κ e S S e . (53)
Using equation (53) and (5), The vector of the elastic forces due to the
transverse deformation is obtained as
T 12
4 20 0
1 2d d
lTt
t t
UEI
f f
Q S S κ S e
e, (54)
wherein f is defined as
1 1
2 T T
0 0
d d elf f r r e S . (55)
In the same manner, an average curvature can be defined as
1 1 1
2 T T T
0 0 0
d d d κ κ r r e S S e . (56)
Substituting equation (55) and (56) into (54), the vector of elastic forces is
defined as
1 2t t t Q K K e , (57)
and the stiffness matrix due to the transverse deformation is obtained explicitely
as follows
2 2
1 3 4
2
12 6 12 6
4 6 21
12 6
. 4
t
l l
l l lEI
ll f
sym l
I I I I
I I IK
I I
I
, (58)
24
2 2
2
2 2
2
6 6
5 10 5 10
2
15 10 3026
5 10
2.
15
t
l l
l l lEI
ll f
lsym
I I I I
I I Iκ
K
I I
I
, (59)
wherein I is 3 by 3 identity matrix. The matrix 1tK is very similar to the
matrix tK of equation (42). 1tK can be siplified to tK , when 1f
which means that the deformation is small. The matrix 2tK contains the terms
related to the derivative /f e . 2tK becomes null matrix when the
deformation is small due to the average curvature 0κ . T2 model for the
elastic forces due to transverse deformations should be used with L2 model for
the longitudinal deformation to study the general case of large deformations. In
order to meet the assumption of constant longitudinal deformation within the
finite element, a sufficient number of element should also be used.
25
2.3. Equations of the motion of a plate element
using ANCF
This chapter derives equations of the motion of a plate element using ANCF
suggested by. Dmitrochenko et al. (2003) and Yoo et al. (2004) based on
Kirchhoff plate theoy with nonlinear strain-displacement relationshops to
obtain elastic forces.
2.3.1. Kinematic description of a plate element
Fig. 7 An arbitrary position vector on the plate element surface
Considering a plate element of size a b h (lengthwidththickness)
as Fig. 7, whose surface is parameterized by value 1p and 2p . An arbitrary
position vector on the plate surface r can be considered as a point on 1 ,
2 beam which is parrarel to the 1p axis and its slopes on the nodes are 1
26
and 2 . From the equation (3), the position vector r can be writtens as
1
11 2 1 2 3 4
2
2
ˆ ˆ ˆ ˆ,p p s s s s
ρ
τr I I I I
ρ
τ
, (60)
wherein 1ˆ ,k ks s p a are Hermite shape fuctions from equation (4), and I
is 3 by 3 identity matrix.
Fig. 8 Nodal coordinate vectors of a ANCF plate element
To determine 1 , 2 , 1 , and 2 for representing an arbitrary vector r ,
16 Degrees of freedom plate element are used as Fig. 8. Each node has 4 vectors
which represent displacement of the node, 1p axis direction slope, 2p axis
direction slope, and second order slope which represent the change of slope
through the other axis. Equation (61) shows the notation of the nodal coordinate
vectors.
27
1
2
1 2
i jijuv i j
p up v
p p
rr (61)
From the notation of the vector ijuvr , superscript i and j represent
derivatives with respect to 1p and 2p parameter, and the subscrip u and v
represent the parameter coordinate on the 1p and 2p axis. For example, 0000r ,
superscript i and j are both zero which means that this vector represent
displacement, and subscript u and v are both zero which means the
coordinate of the node is 1 2( , ) ( , ) (0,0)p p u v . In the same manner, 1010r
represent the 1p axis slope vector on the nodal point 1 2( , ) (1,0)p p . With
this notation, we can determine 1 , 2 , 1 , and 2 for representing an
arbitrary vector on the plate surface from equation (60). Fig. 9 shows how to
represent 1 with nodal vectors.
Fig. 9 Nodal coordinates to determine 1 on 2p parameter beam.
To figure out 1 , let us assume a 2p beam which include point 1 and
its nodal coordinates are 00 01 00 0100 00 0 0b b e r r r r . Using equation (3),
28
position vector of 1 can be written as
00000100
1 1 2 3 4 000010
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆb
b
s s s s
r
rρ I I I I
r
r
, (62)
wherein 2ˆ̂ ,ks s p b are Hermite shape fuction from equation (4), and I
is 3 by 3 identity matrix.
Fig. 10 Nodal coordinates to determine 1 on 2p parameter beam.
To determine 1 which is the slope pararell to the 1p axis on 1 , 1p
axis direction slope vectors on the nodes are used as displacement and second
order slopes are used as slope vector on beam. Therefore, nodal coordinates are
10 11 10 1100 00 0 0b b e r r r r and using equation (3), position vector of 1 can
be written as
10001100
1 1 2 3 4 100110
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆb
b
s s s s
r
rτ I I I I
r
r
. (63)
29
Fig. 11 Nodal coordinates to determine 2 on 2p parameter beam.
Fig. 12 Nodal coordinates to determine 2 on 2p parameter beam.
The same manners are used to determine 2 and 2 as Fig. 11 and Fig.
12, and it can be written as follows.
000
010
2 1 2 3 4 000
01
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ
a
a
a
ab
s s s s
r
rρ I I I I
r
r
(64)
30
100
110
2 1 2 3 4 100
11
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ
a
a
a
ab
s s s s
r
rτ I I I I
r
r
(65)
Finally, substituting equation (62), (63), (64) and (65) into equation (60), a
shape fucntion of the plate becomes equation (66), and nodal coordinate is
equation (67)
11 12 13 14 41 42 43 44; ; ;S S S S S S S SS I I I I I I I I (66
)
00 01 00 01 10 11 10 11 00 01 00 01 10 11 10 1100 00 0 0 00 00 0 0 0 0 0 0
T
11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44
b b b b a a ab ab a a ab ab
e r r r r r r r r r r r r r r r r
e e e e e e e e e e e e e e e e
(67
)
wherein ijS is defined as
1 2ˆˆ ˆ, ,ij i j i jS s p a s p b s s . (68)
Therefore, the shape function of a plate S is 3 by 48 matrix, and the nodal
coordinate e is 48 by 1 matrix.
2.3.2. Mass matrix of a plate element
The mass matrix of a plate element is derived from a kinetic energy of a
plate element like a beam element. A kinetic energy of a plate element is
defined as
T T
1 2
0 0
1 1d d
2 2
a b
T p p r r e Me , (69)
31
wherein is a suface density of the plate, a mass matrix M is defined as
T d
P
P M S S . (70)
The integration operator dP
P is defined as 1 2
0 0
d da b
p p . Due to the size
of the shape matrix S , the sized of the mass matrix M is 48 by 48, from this
large size of the matrix, an explicit expression for the mass matrix can be
obtained in block matrix form as Euqation (71) using definition of the shape
function matrix which is equation (66).
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
M M M M
M M M MM
M M M M
M M M M
, (71)
wherein each block ijM is also defined as block matrix as
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
ij ij ij ij
ij ij ij ij
ijij ij ij ij
ij ij ij ij
M M M M
M M M MM
M M M M
M M M M
, (72)
and ijklM is defined as
ijkl ijklMM I , (73)
wherein ijklM is defined as equation (74) using the definition of the shape
functions.
32
1 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆ ˆˆ ˆ ˆ ˆd d
ijkl ik jl i k j lP P
a b
i j k l
M S S P s s s s P
s s p s s p
(74)
2.3.3. Stiffness matrix of a plate element
Stiffness matrix is derived from the strain energy of a plate, and this ANCF
plate element uses Kirchhoff plate theory to derive strain energy. Following
Kirchhoff plate theory, the strain energy can be decamped into two components
that one due to longitudinal and shear deformation in the midplane, and the
other one due to its bending and twist. Therefore, the strain energy of the plate
can be written as
U U U . (75)
U is the strain energy due to longitudinal and shear deformation which is
defined as
2 2
2 1122 11 222
1 1
62 dij ij
i jP
U D D Ph
, (76)
and U is the strain energy due to bending and twist which is defined as
2 2
2 1122 11 22
1 1
12 d
2 ij iji jP
U D D P
, (77)
wherein ijD is flexural rigidities defined as
33
3
12 21
3
12 1
6
ijij
ijij
E hD i j
E hD i j
, (78)
and additional stiffness 1122D is defined as
1122 11 21 22 120.5D D D . (79)
ijE is Young’s modulus when i j and shear modulus when i j . ij
is Poisson ratio of the plate. ij is Green-Lagrange strain tensor defined as
T1
2ij i j ij r r , (80)
wherein ij is Kronecker delta defined as
1
0
ij i j
i j
. (81)
ij is the transverse curvature when i j , and the twist curvature when
i j which are calculated by equation (82)
T
3
ijij
r n
n, (82)
wherein the normal vector n is defined as
1 2 n r r , (83)
and the derivative of the position vector ir and ijr are defined as
i
i mn mni
Sp
rr e ,
2ij
ij mn mni j
Sp p
rr e , (84)
34
wherein iS and ijS are the derivatives of the shape function that are defined
as follows.
iklkl
i
SS
p
,
2ijklkl
i j
SS
p p
(85)
(1) Midplane elastic forces model
The midplane elastic forces are gradient vectors of the midplane strain
energy which defined in equation (76). Therefore, midplane elastic forces are
calculated by equation (86).
11 11 2222 22 112
12dij
kl ij ijPkl kl kl kl
UD D P
h
Qe e e e
, (86)
Longitudinal and shear deformation tha defined by equation (80) and their
gradient can be written as equation (87) and (88) in terms of shape functions
and nodal coordinates
T1
2i j
ij mn pq ijS e e , (87)
ij i j
klrs rskl
S
ee
, (88)
wherein i jmnpqS is defined as follows.
1
2i j i j j imnpq mn pq mn pqS S S S S , (89)
① L1 model
L1 model considered simplified midplane force model. The simplification
consists in assumption that longitudinal and shear deformations are constan
35
over the plate element and approximated by equation (90).
T T T T
00 00 00 0011 0 00 0
00 00 00 0022 0 00 0
10 01 10 01 10 01 10 0112 21 00 00 0 0 0 0
1/ / 1
21
/ b / 12
1
8
a ab b
b ab a
a a b b ab ab
a a
b
r r r r
r r r r
r r r r r r r r
, (90)
To derive stiffness matrix of the L1 model, substituting equation (88) and
(90) into equation (86), and it becomes the equation as follows
kl klmn mn Q K e , (91)
with the stiffness matrix klmnK that defined as
11 1 1 2 222 22 112
12 i jklmn ij ij klmn klmn klmnD S D S S
h K I , (92)
wherein i jmnpqS
is the plate integration of i jmnpqS that defined as
1d
2i j i j j imnpq mn pq mn pq
P
S S S S S P . (93)
The explicit expression for i jmnpqS
is in Appendix. In equation (92) the
deformations ij are constant and equation (91) is linear in number of the
element nodal coordinate e . Therefore, this model is called L1
② L3 model
L3 model has no simplifications are used for the longitudinal and shear
deformations. In L3 model, elastic forces are cubic in e . That is the reason
why this model called L3.
36
To evaluating the stiffness matrix of L3 model, equation (87) and (88) are
substituted into equation (86), and the stiffness matrix can be written as
Tpqrsklmn klmn pq rs klmnK K K e e I , (94)
wherein mpqnklrsK and klmnK are defined as follows.
; 11 1 1;2 2 2 2;1 1; 22 ; ;2
6mpqn i j i jklrs ij klrs mnpq klrs mnpq klrs mnpqK D S D S S
h (95)
11222
6 i jklrs ij ij klrsK D D S
h (96)
The symbols i jklrsS and ;
;i j i jklrs mnpqS are defined as
i j i j
klrs krlsPS S dP
, ;
; di j i j i j i j
klrs mnpq krls mnpqPS S S P
. (97)
Exiplicit expression of ;;
i j i jklrs mnpqS is in Appendix.
(2) Transverse elastic forces model
The gradients of the strain energy due to transverse deformation which is
equation (77) become the transverse elastic forces as
11 22 1122 11 22 dij
kl ij ijkl kl kl klP
UQ D D P
e e e e
. (98)
Curvatures of the plate element defined in equation (82) can be written as
T
3
ijij f
r n
, Tf n n n , (99)
and its derivatives are defined as
37
T TT
T3 T 4
1 3ij ijij ij
kl kl kl kl
f
f f
r nn r r n
e e e e, (100)
wherein kl
fe
can be calculated as follows.
T
T
1
kl kl
f
f
nn
e e (101)
① T2 model
T2 model is the simplified model as L1 model which describing longitudinal
and shear deformation. Like L1, T2 model uses simplified curvatures to
calculate transverse deformation of the plate. This simplified curvatures are
calculated by implementing the idea of the average normal vector which is the
mean value of the normal vectors at the corners of the plate as
21 12 41 32 23 14 43 34
1
4 n e e e e e e e e . (102)
From equation (102), f in equation (99) becomes f as
Tf n n n . (103)
Then the derivatives of equation (100) become
T
T ij kl ijkl
nr b r
e (104)
and equation (101) becomes as
1
klkl
f
f
b n
e, (105)
38
wherein klb is defined as follows.
1, 1
1, 1
1, 2,1 , 4,1 , 2,3 , 4,3
41
, 1, 2 , 3, 2 , 1, 4 , 3, 44
0 ,
k l
kl k l
for k l
for k l
for the rest combinations of k l
e
b e (106)
Finally, derivatives of the curvatures become as
T3 5
1 3ij ij ij ijkl mn kl mn mn mn kl
kl
S S Sf f
n b e e n n b
e (107)
Substitution equation (104), (105), and (107) into equation (98), the elastic
forces due to the transverse deformation becomes as
T T3 2
1 3kl pq klpq klpq kl mn pq klpq klQ S S S
f f
e n n b e e n n b , (108)
wherein mnpqS is defined as
11 1122 221122
ijijklpq ij klpq klpq klpqS D S D S S , (109)
and ijijmnpqS defined as
dijij ij ijmnpq mn pq
P
S S S P . (110)
Equation (108) is quadratic in the nodal coordinates e . Therefore, this
model called T2. If the deformation is small enough, equation (108) can be
simplified by choosing first term as follows.
T3
1kl pq klpqQ S
f e n n (111)
39
② T3 model
T3 model uses the normal vector which is averaged over the whole plate
surface using double integration. This normal vector is defined as
1
dmnpq mn pq
P
S Pab
n e e , Tf n n n , (112)
wherein mnpqS is defined as
1 2 2 11
2mnpq mn pq mn pqS S S S S . (113)
Then, the derivative terms in equation (100) become as follows.
T
T ij klmn mn ijkl
S
nr e r
e (114)
klmn mnkl
fS
e ne
(115)
Finally, derivatives of the curvatures become as
T3 5
1 3ij ij ij ijkl klmn klmn mn mn klmn mn mn pq
kl
S S S S S Sf f
e e n n ee
. (116)
Therefore, substituting equation (114), (115), and (116) into equation (98),
The elastic forces due to the transverse deformation is obtained as
T T6 2
1 3kl rs klmnpqrs mnpqklrs mn klpqmnrs mn pqQ S S S
f f
e n e e n n e , (117)
wherein the symbol klmnpqrsS is defined as
11 1122 221122
ijijklmnpqrs ij klmnpqrs klmnpqrs klmnpqrsS D S D S S , (118)
and the symbol ijijklmnpqrsS is defined as
40
dijij ij ijklmnpqrs klmn pq rs
P
S S S S P . (119)
Therfore, T3 model of elastic forces is cubic in e , that’s why this model
called T3.
Explicit expression of the symbols are in Appendix.
41
2.4. Derivation of the general form of the
equations of the motion of the flexible
multibody system
The equations of the motion of the flexible mutibody system are derived
from Euler-Lagrange equations which defined as
Td T T U W
dt
qCq q q q . (120)
wherein T is the kinetic energy of the system, U is the potential energy of
the system, W is the virtual work due to the external forces, and q is the
generalized coordinates of the system. qC is the constraint Jacobian matrix
which represents kinematic constraint which called joint that will be considered
in following chapter. is the vector of Lagrange multipliers and the term
TC represents the constraint forces of the system. The first two terms in
equation (120) can be written as
T
T1
2
d T T
dt
Mq Mq q Mq
q q q
. (121)
However, in ANCF system
T
T1
2
Mq q Mq
q is zero, because the
mass matrix is a constant matrix shown by preceding chapters. The derivative
of the potential energy can be written as,
U
Kqq
. (122)
wherein, K is the stiffness matrix which was shown by preceding chapters.
Therfore, equation (120) becomes as,
42
T
e qMq Kq C Q . (123)
wherein eQ is the generalized force due to the external forces that defined as
e
W
. (124)
The generalized force eQ is easily calculated by the multiplying shape
functions and the external forces. For instance, if the external force eF is
point load, eQ is calculated as
Te
e Q F S , (125)
wherein S is the shape function of the element, and if eF is uniform
distributed load, then eQ is calculated as
T T
de ee
V
V Q F S F S . (126)
Explicit expression for S of a beam element is shown as
1 1
2 12 2 12
l l S I I I I , (127)
and S of a plate element is shown as
1 1
4 24 4 24 24 144 24 144
1 1
4 24 4 24 24 144 24 144
b b a ab a ab
b b a ab a ab
S I I I I I I I I
I I I I I I I I, (128)
Wherein I is 3 by 3 identity matrix and, a and b are the breadth width of
the plate.
The constraint C is a equation of the generalized coordinates and time and
43
it can be written as
, 0t C q . (129)
The time derivative of equation (129) is derived as
0t qC q C . (130)
where, the subscript represent the derivatives. Differentiating equation (130)
with repect to the time yields
2 0t tt q q q qC q C q C q q C . (131)
Combining equation (123) and (131), the equations of the motion of the
multibody system can be written with matrix form as equation (132) which
called augmented formulation.
q
q 0
Te
c
Q KqM C q
C λ Q
. (132)
wherein cQ is the residual term from the equation (131) that defined as
2 0c t tt q q qQ C q C q q C . (133)
Equation (132) is the form of the Newton’s equation which is Mq F .
44
2.5. Kinematic constraints
In a mechanical system, the bodies may be interconnected by one or more
kinematic constraints which called joints which can be described as algebraic
constraint equations as equation (129) that constrains some degrees of freedom
of the body (Nikravesh, 1988).
Implementing mechanical joints into the equations of the motion that are
expressed by augmented formulation like equation (132), The Jacobian of the
constraint qC and cQ which defined in quation (133) have to be determined.
However, ttC is usually zero when the constraint equation is not function of
the time. Therefore, only the qC and tqC have to be determined.
In following chapters, some types of mechanical joints and their constraint
equations which are used in this study are introduced.
2.5.1. Rigid body – Rigid body constraints
Kinematics of the rigid body are represented by the position of the center of
the gravity of the body, and its orientation. For the orientation expression,
Eulear ange and the Quarternian is broadly used. In this study, the orientation
of the bodies are represented by Euler angle. An arbitrary pont on the rigid body
can be shows as Fig. 13, and it can be written as,
45
Fig. 13 An arbitrary position vector on the rigid body
E E E GP G G P r r R r . (134)
wherein the notation of the vector are defined as
/ : A -referecne frame, B-Designated point, C-Orientation pointAB Cr . (135)
For example, in Fig. 13, the vector /E
P Er represents pointing position of
the P from the point E with respect to the E frame. Another example, the
vector /G
P Gr points the point P from the point G . However this vector is
described with respect to the G frame which is body reference coordinate. A
position of the G is center of gravity of the body. The matrix ABR is the
rotation transformation matrix that transformation the coordinate system from
the B frame to the A frame. In the 3 Dimensional space, EGR is defined
as
cos cos cos sin sin sin cos cos sin cos sin sin
cos sin cos cos sin sin sin cos cos cos sin cos
sin sin sin cos cos
EG
R . (136)
46
wherein , , and is Euler angles that defined as
Fig. 14 Definition of Euler angles
A rigid body without constraints has 6 degree of freedom, 3 for the
translational motion, and 3 for the rotational motion, and the constraints reduces
the degree of the freedom of the body. The objectives of evaluating constraint
force are get the constraint equation which defined as equation (129), and
deriving the qC and tqC from equation (129) for substituting the terms into
the equation of the motion as equation (132). If there are 2 rigid bodies 1G
and 2G , the generalized coordinates q in equation (129) is defined as
1 2
1 1 2 2
TG GE EG G G G
q r θ r θ . (137)
wherein 1
EGr and
2
EGr are the position vector of the center of gravity of the
bodies, and 1
1
GGθ and 2
2
GGθ are the orientations of the bodies which is
described by Euler angle.
47
(1) Ball joint
Fig. 15 Kinematics of a ball joint
A ball joint is the joint that restrains three translation degree of freedom of
the bodies as Fig. 15. Therefore, The joint position 1GPr and 2G
Pr point out
same position through out the time. Therefore the constraint equation of the ball
joint is defined as
1 2
1 1 1 2 2 2
( ,Tr3)/ / / / 0G GBall E E E E
G E G P G G E G P GR R C r r r r . (138)
The time derivative of the equation (138) becomes as
1 2
1 1 1 1 2 2 2 2
1
11 2
1 1 2 2
2
2
( ,Tr3)
/ / / /
/
/
/ /
/
/
0Ball
G GE E E E E EG E G G P G G E G G P G
EG E
EG EG GE E
G P G G P G EG E
EG E
d
dt
q
Cr R r r R r
r
I R r I R r C qr
. (139)
wherein qC of the ball joint is defined as
48
1 2
1 1 2 2
( ,3)
/ /
trG GBall E E
G P G G P GR R q
CC I r I r
q, (140)
and the tilde a represent the cross product of the vectors into the matrix form
that defined as
1 2
1 3
2 3
0
0
0
a a
a a
a a
a when, 1
2
3
a
a
a
a , (141)
which satisfies a identity as
1
2
3
a
a
a
a ,1
2
3
b
b
b
b ,
2 3 3 2
3 1 1 3
1 2 2 1
a b a b
a b a b
a b a b
a b ab .
(142)
tqC of the ball joint is defined as
1 2
1 1 1 2 2 2/ /G GBall E E E E
t G G P G G G P GR R qC I r I r , (143)
wherein EG is the global anglular velocity of the body and I is 3 by 3
identity matrix.
49
(2) Cylindrical joint
Fig. 16 Kinematics of a cylindrical joint
Cylindrical joint restrains two translation, and two rotational degree of
freedom of the body. A body attached on the cylindrical joint can move just
follow the joint axis, and it can only rotate with a joint axis. To determine
kinematics of the cylindrical joint, we assume 4 points on the joint axis. The
point 1G and 1Q are attached to the body 1, and 2G and 2Q are attached
to the body 2. The vector 1s is the vector from the point 1G to 1Q , and 2s
is the vector from the point 2G to the 2Q , and The vector d is the vector
from the point 1Q to 2Q as Fig. 16. To restrain rotation, The vector 1s and
2s are always parallel to each other, and to restrain translation motion, vector
1s and d are also parallel to each other. From this identities, the constraint
equations of the cylindraical joint are defined as,
(Cylindrical, ,2)1 1 0tr C s d s d
(Cylindrical, ,2)1 2 1 2 0ro C s s s s .
(144)
50
To obtain qC of the cylindrical joint, same manners are used like equation
(140), and the qC of the cylindrical joint is defined as
1 1 1 1 2 2 2 1 1 1 1
1 1 1
1 1 1 1 1 1
2 2 2 1 1 1
1 1 1 1 1 1 2 2 2
1 1 1 2 2 2
( ,2)
/ / /
/
/ /( ,2)
/ /
/ / /
/ /
0
0
tr E E E E EG G S G G G S G G Q GE
G Q GE E
G Q G G S Gro
E EG Q G G Q G
E E EG Q G G Q G G S G
E EG Q G G Q G
r R r R RR
R R
R R
R R R
R R
q
C r r rrq
C r rC
r rq
r r r
r r
. (145)
tqC of the cylindrical joint is defined as
1 1 1 1 1 2 2 2 2 1
1 1 1 1 1 1 1 2 2 2 1
1 1 1 1
1 1 1 1 1 1 1 1 1 1 1
1 1 1 2 1 1 1
2 2 2 1
/ /
/ / /
/./ / /
/ /
2 /0
E E E E E EG G G S G G G G S G
E E E E EG Q G G G S G G G S GE E
G G Q Gcylind E E E E Et G G Q G G G Q G G S G
E E EG Q G G G S G
E E EG G Q G G Q
q
r ω R r r ω R r
R r r R r r R rω R r
C ω R r ω R r R r
R r ω R r
ω R r R r
1 1 2 2 2 1 1 1 1
1 1 1 1 1 1 1 1 2 2 2 1 1 1 2 2 2 2
1 1 1 1 2 2 2 1 1 1 2 2 2 2
/ / /
/ / / / /
/ / / /0
E E EG G Q G G G Q G
E E E E E E E EG G Q G G G Q G G S G G Q G G G S G
E E E E E EG G Q G G Q G G Q G G G Q G
R r ω R r
ω R r ω R r R r R r ω R r
ω R r R r R r ω R r
.
(146)
qC and tqC have 6 rows. However, cylindrical joints retrain 4 degree of
freedom, by that means, trqC and ro
qC has dependent components. Therefore,
Two independent components have to be choosed when substituting equation
(145) and (146) into equation (132).
51
(3) Fixed joint
Fig. 17 Kinematics of a fixed joint
Fixed joint restrains full degree of freedom of the body that are three
translation, and three rotational degree of freedom. A fixed joint, name it self,
fix two bodies in a point. Therefore, the position vector that designates point
P has to be same with respect to body 1 and body 2, and the difference of the
initial orientation of the bodies is to be same through out the time. From this
characteristic, the constraint of a fixed joint as defined as
1 2
1 1 1 2 2 2
( , r,3)/ / / / 0G GFix t E E E E
G E G P G G E G P GR R C r r r r .
( ,rot,3) 0 01 2 1 2 0Fix C θ θ θ θ ,
(147)
wherein θ denotes the orientation of the body. the superscript 0 means at the
initial point, and subscript means the body.
qC and tqC of the fixed joint is derived from equation (147), and they are
as
52
1 1 2 2
( ,3)
( ,6) / /
( ,3)0 0
tr
E EFix G P G G P G
ro
R R
q
C
q I r I rC
C I Iq
, (148)
1 1 1 2 2 2( ,6) / /0 0
0 0 0 0
E E E EFix G G P G G G P G
t
R R
q
r rC , (149)
2.5.2. ANCF beam – ANCF beam constraints
Fig. 18 Kinematic description of an ANCF beam element (Fig. 5)
The kinematic description of an ANCF beam element has already explained
in the chapter 2.2.1. Recall that, an ANCF body already has its position vector
of the nodes as nodal coordinates in absolute coordinate that makes constraints
simple. However, Orientation of the ANCF beam is hard to be shown, because
the shape of the beam is represented by the slopes of the nodes. Recall that the
kinematic description of i -th body’s ANCF beam can be written as follows.
0
01 2 3 4
i
ii
ili
l
p s s s s
r
rr I I I I
r
r
, (150)
53
(1) Ball joint
Like a ball joint of rigid bodies, a ball joint of ANCF beams restrains three
translation degree of freedom of connected body. Assuming ANCF beam i
and beam j that are connected on the second node of the beam i and the
first node of the beam j , constraint equation can be written as
( , r,3)0 0Ball t i j
l C r r , (151)
wherein the generalized coordinates q are defined as
T
0 0 0 0i i i i j i j i
l l l l q r r r r r r r r . (152)
Therefore, qC is calculated as
( ,3)
0 0 0 0 0 0tr
Ball
q
CC I I
q, (153)
and tqC is zero matrix due to equation (153) is constant matrix.
(2) Fixed joint
A fixed joint of an ANCF beam retrains three translation and rotaion degree
of freedom of the bodies connected with a fixed joint. However, rotation
parameters of an ANCF beam are not shown explicitly but represented by its
slope vector that represent tangential direction of the beam. If a fixed joint
connects two ANCF beam, the joint restrains rotational degree of freedom by
constraining slopes of the beam. Assuming ANCF beam i and beam j that
are connected connected on the second node of the beam i and the first node
of the beam j , constraint equation can be written as
54
( , r,3)0 0Fix t i j
l C r r
( , ,3)0 0Fix ro i j
l C r r , (154)
Therefore, qC is calculated as
( ,3)
( ,3)
0 0 0 0 0 0
0 0 0 0 0 0
tr
Fix
ro
q
C
I IqC
I IC
q
, (155)
and tqC is also zero matrix due to equation (155) is constant matrix.
2.5.3. ANCF beam – Rigid body constraints
Fig. 19 Kinematics of an ANCF beam and a rigid body
In preceding chapers, the joint between rigid-rigid bodies and ANCF beam-
rigid body have explained. In this chapter, joint between ANCF beam and rigid
body will explaned. If you need details about general ANCF constraint
formulation, Sugiyama (2003) derived general form of the ANCF constraints
equations.
55
Kinematic descriptions of an ANCF beam and a rigid body are quite
different, especially representing orientation of the body, because orientation
of the rigid body is represented by its Euler angle, but the ANCF beam is
represented by its slopes at the nodes. Due to this reason, rotational constraints
between an ANCF body and a rigid body have to be represented with relation
ships of the slope of the ANCF body and an aribitrary vector which is attached
to the rigid body, and its initial direction is same to the slope of the ANCF body.
The generalized vector of the constraints defined as
0 0i i i i E E
l l G G q r r r r r . (156)
(1) Ball joint
A ball joint restrains three translation degree of freedom of the body, and it
is easy to derive constraint equation of the ball joint between an ANCF body
and a rigid body. Assuming the ANCF beam i and the rigid body G are
attached on the point P with a ball joint as Fig. 19. Constraint equation is
obtained as
2( , r,3)/ / 0GBall t i E E
l G E G P GR C r r r , (157)
Therefore, qC is calculated as
( ,3)
/0 0 0tr
Ball EG P GR
q
CC I I r
q, (158)
and tqC is defined as
2/0 0 0 0 GBall E E
t G G P GR qC I r . (159)
56
(2) Fixed joint
Fig. 20 Kinematics of a fixed joint between an ANCF beam and a rigid body
A fixed joint restrains three translation and three rotational degree of
freedom of the body. Let us Asumme that a ball joint attached on the point P
as Fig. 20. lr , the slope at the node P , is the direction of the beam at the node.
Constraints that restrain translation motion is same as ball joint. However, to
restrain rotation degree of freedom, the vector /Q Pr which is parallel to lr
initially and attached on the rigid body G has to be defined. From the vector
/Q Pr , orthogonal vectors 1/N Pr and 2/N Pr that are attached to the body are
defined as
1/P / /N Q P C P r r r ,
2/P / 1/N Q P N P r r r , (160)
wherein the /C Pr is an arbitrary vector that is not parallel to /Q Pr .
The slope vector lr and the vector /Q Pr that on the body have to be
parallel each other through out the time. Therefore, constraint equations are
defined as
2( , r,3)/ / 0GFix t i E E
l G E G P GR C r r r , (161)
57
( , ,1) E GG N1/Pˆ 0Fix ro
l C r R r ,
( , ,1) E GG N2/Pˆ 0Fix ro
l C r R r ,
wherein ˆr is the unit vector of the r defined as
Tˆ
rr
r r. (162)
Equation (161) restrains only two rotational degree of freedom due to the
ANCF beam model represent only direction of the beam centerline. Therefore,
the rotation through the centerline cannot be included in the degree of freedom
of the ANCF beam.
From the constrain equation (161) qC is calculated as
( ,3)
( ,2)
tr
Fix
ro
q
C
qC
C
q
(163)
( ,3)E G
G P/G0 0 0tr
CI I R r
q (164)
T
T(ro,1)TE G E G
G N1/P G N1/P
10 0 0 0
l
TT Tl ll l l l
r rC rR r I R r
q r rr r r r
. (165)
T
T(ro,1)TE G E G
G N2/P G N2/P
10 0 0 0
l
TT Tl ll l l l
r rC rR r I R r
q r rr r r r
. (166)
tqC is calculated as
( ,3)
( ,2)
d
d
tr
Fixt ro
t
t
q
q
q
C
CC
(167)
58
( ,3)
E GG N2/P0 0 0 0 0
d
trE
Gt
qC
ω R r (168)
T
T(ro,1)TE G E G
G N/P G N/P
d d 1 d0 0 0 0
d d dl
TT Tl ll l l l
t t t
qr rC r
R r I R rr rr r r r
(169)
wherein
TTE G
G N/P
d 1
dl
TTl ll l
t
r rR r I
r rr r is,
TTE E G
G G N/P TT
TTE G
G N/P TT
1
1
l l
l ll l
l l
l l ll lt
r rω R r I
r rr r
r r rR r I
r r rr r
,
T
TT
T T
T T T3/2 TT
1
31
l l l
l l ll l
l l l l
l l l l l l
l ll l
t
r r rI
r r rr r
r r r rr r r r r r I
r rr r
,
(170)
and T
E GG N/P
d
d Tl l
t
rR r
r r is
T
E GG N/P
TT
E GG N1/P
T
E E GG G N1/P
d
d
1
l
Tl l
l l
lTTl ll l
l
Tl l
t
rR r
r r
r rI r R r
r rr r
rω R r
r r
. (171)
59
2.6. External forces
The hydrostatic force, hydrodynamic force, gravity, and current force are
applied as external forces in the motion analysis of the drilling rig with a riser.
2.6.1. Hydrostatic force
The hydrostatic force is acting on the bodies that are floating on the surface
of the water or submerged in the water. The hydrostatic forces on the hull are
calculated as
hydrostatic SW platformg F , (172)
wherein SW is the density of sea water, g is the acceleration of gravity,
which is 9.81 m/s2, and platform is the displacement volume of the patform.
The kernel (Cha et al., 2010) calculates the displacement volume in each time
step to calculate the hydrostatic forces.
2.6.2. Hydrodynamic force
Hydrodynamic forces can be divided into the wave-exciting force and the
radiation force, as shown in equation (173). The wave-exciting force is exerted
by the incident wave, and the radiation force is generated by the motion of the
floater itself.
hydrodynamic exciting radiation F F F (173)
excitingF is obtained from the force RAO (Response Amplitude Operator) and
60
the sinusoidal function at a given frequency. The force RAO can be obtained
from a commercial solver, such as WADAM by DNV (Det Norske Veritas,
2002). To calculate radiationF in the time domain, the Cummins equation can be
used (Cummins, 1962). The added mass ija and the damping coefficient
ijb can also be obtained from the commercial solver. Fig. 21 shows the
calculation procedure.
Fig. 21 Calculation procedure of hydrodynamic forces
For the simulation, the hydrodynamic forces are pre-calculated with the
commercial hydrodynamics tool WADAM by DNV in the simulated draft (Det
Norske Veritas, 2002). WADAM uses a 3D panel method to evaluate velocity
potentials and hydrodynamic coefficients. The radiation and diffraction
velocity potentials on the wet part of the body surface are determined from the
solution of an integral equation obtained by using Green’s theorem with the free
surface source potentials as the Green’s functions. The source strengths are
evaluated based on the source distribution method using the same source
potentials. The velocity potential is the sum of incident wave potential,
diffraction velocity potential, and radiation velocity potential. Hydrodynamic
pressure on the hull are determined from the Bernoulli’s equation, and the
hydrodynamic force is obtained from the result of the surface integral of the
hydrodynamic pressure. Hydrodynamic force consist of Froude-Krylov force,
diffraction force, and the radiation force which are obtained from incident wave
61
potential, diffraction velocity potential, and radiation velocity potential. From
the result of calculation of the radiation force, the added mass coefficient and
the damping coefficient are obtained, and Froude-Krylov force and diffraction
force become wave exciting force.
2.6.3. Gravity on the ANCF body
The gravity of the body is defined as
0
0gravity
mg
F , (174)
wherein m is the mass of the body and g is the acceleration of gravity
which is 9.81 m/s2. However, to calculating gravity acting on an ANCF body,
generalized elastic force due to the gravity has to be defeind using equation
(126). Therefore, gravity acting on on ANCF body can be written as
T T
,
0
d 0e ee gravity
V
V
mg
Q F S F S S , (175)
because the gravity is acting like uniform distributed load.
62
2.6.4. Current force on the ANCF body
The force due to the current is calculated from the Morison equation. The
Morison equation is sum of two force components which are inertia force and
drage force. The Morison equation is defined as
1
2current m sw dC V C A F u u u , (176)
wherein is the density of the fluid mC is inertia coefficient, dC is drag
coefficient, V is the volume of the body, A is the cross section area of the
body perpendicular to the flow direction, and u and u are the velocity and
the acceleration of the fluid. The inertia force in the Morison equation is the
term of the acceleration of the fluid, and the drage force is the term of the square
of the fluid velocity. If the body is ANCF body, Generalized elastic force due
to the current has to be defeind using equation (126). Therefore, the current
force acting on ANCF body can be written as
T T
, de current current current
V
V Q F S F S . (177)
63
2.7. Method of numerical integration
The 4th order Runge-Kutta method is used for the numerical integration to
get the velocities and the displacements from the accellerations that are solution
of the equation (132).
If there is a initial value problem as
,y f t y , 0 0( )y t y , (178)
wherein y is unknown functions of time t which we would like to
approximate.
The solution of the 1n -th time step is defined with the function value of
n -th time, and with time step h as
1 1 2 3 4
1
2 26n n
n n
hy y k k k k
t t h
, 0, 1, 2,3,n
where,
1
2 1
3 2
4 3
, ,
, ,2 2
, ,2 2
, .
n n
n n
n n
n n
k f t y
h hk f t y k
h hk f t y k
k f t h y hk
(179)
1k is the increment based on the slope at the beginning of the interval using
y , 2k is the increment based on the slope at the midpoint of the interval using
12
hy k , 3k is again the increment based on the slope at the midpoint, but
64
using 22
hy k , and 4k is the increment based on the slope at the end of the
interval using 3y hk . In averaging the four increments, greater weight is
given to the increments at the midpoint. The 4th order means that the local
truncation error in on the order of 5O h , while the total accumulated error is
order 4O h .
65
2.8. Configurations of the simulation program
Fig. 22 Configurations of the simulation program
To implement the flexible multibody dynacmis code, we developed the
prototype program as Fig. 22. This prototype program consists with 5 library
modules that are MBD body library, MBD joint library, MBD foce library,
MBD solver library, and 3D visualization library. MBD body library is the
library for modeling rigid and flexible body to build equations of the motion of
the multibody system. This library constains set of the equations of the motion
of the bodies that are formulated automatically by body parameters. MBD joint
library contains the equations of the joints of the multibody system which can
apply to rigid bodies and flexible bodies. MBD force library is the module for
the calculating external forces of the multibody system such as gravity,
hydrostatic, hydrodynamic, current, and spring-damping forces. MBD solver
library has varies of numerical integrator, and inverse matrix calculator such as
4th order Runge-Kutta, Euler method for numerical integrator, and LU
factorization, Gaussian elimination, and Cholesky factorization for inverse
matrix calculation. 3D visualization library visualize the result of the system.
66
A cormmercial visualization code HOOPS 3D is used for this library.
Fig. 23 Simulation sequence of the anlaysis program
The simulation sequence of the analysis promgram is shown as Fig. 23. First,
the mass matrices of the bodies are calculated from the body information, and
joint constraints, external forces, and elastic forces from the flexible bodies are
calculated. When augmented formulation has formulated, Inverse matrix
calculation of the augmented formulation has performed to get accelerations of
the system. The next step is time integration that calculates velocities and the
positions of the system. The 4th order Runge-Kutta is used for time integration.
After obtaining positions, and velocities, the simulation kernel visualize the
result in 3 dimensional graphic, and advances time step. Finally, new external
forces are calculated with previous step’s information and formulates new
equations of motion in advanced time step.
67
Motion analysis of a drilling rig
with a riser
This chapter shows the verification of the flexible mutibody dynamics code
and the modeling of the main examples and their results. For the verification of
the flexible multibody dynamics code, comparison studies of the numerical
analysis and the analytic solution of the beam element and plate element have
been shown, and comparison study of the commercial code has been performed.
Modeling of the motion analysis of a drilling rig with a riser is described in this
chapter. First, configurations of the drilling rig is shown, and modeling of a
wire-line riser tensioner is described. Finally, the simulation results of the
simulation are shown in the end.
3.1. Verification of flexible elements
To verify the flexible multibody dynamics code, comparison study between
numerical analysis and the analytic solution of the beam and plate element have
been performed.
3.1.1. Verification of a beam element
The analytic solution for the deformation of a 2D cantilever beam is used to
valify the numerical beam model. The specifications of the 2D beam are shown
in Table 3; the configuration of the model is shown in Fig. 24; and the result of
the numerical model is shown in Fig. 25.
68
Table 3 Specifications of a 2D cantilever beam
Item Value
Length 10 m
Mass 795 kg
Young’s modulus 210 GPa
Second moment of inertia 8.33 x 10-6 m4
Density 7,850 kg/m3
Area 0.01 m3
No. of elements 1~10
Fig. 24 Deformation of the 2D cantilever beam
Fig. 25 Convergence test of the numerical cantilever beam model
The analytic solution of a 2D cantilever beam is as follows (Gere and
Timoshenko, 1997):
69
4
8b
qL
EI , (180)
wherein q is the uniform distributed load, L is the length of the beam, E is
Young’s modulus, and I is the second moment of inertia of the beam. The
analytic solution is 0.0557 m, and the result of the numerical analysis is
converged to the analytic solution when the number of element increases. This
shows the flexible dynamics code for the beam is reliable.
3.1.2. Verification of a plate element
The analytic solution for the deformation of a pure bending plate is used to
valify the numerical plate model. The specifications of the plate are shown in
Table 4; the configuration of the model is shown in Fig. 26; and the result of
the numerical model is shown in Fig. 27.
Table 4 Specifications of a plate element
Item Value
Length a, b 10 m
Thickness 0.1 m
Density 7,950 kg/m3
Young’s modulus 210 GPa
Poison ratio 0.3
Uniform load 7,798.95 Pa
No. of elements 12~62
70
Fig. 26 Dimension of a plate element
Fig. 27 Convergence test of the pure bending of numerical plate model
The analytic solution of a pure bending of a plate is as follows (Timoshenko
and Woinowsky-Kriger, 1959):
2
0
6 3 2 21 1
2 2
sin sin4 1
3 m n
m x n yq a bw
Eh m nmn
a b
, (181)
71
wherein 0q is the uniform distributed load, a , b , h are the dimensions of
the plate, E is Young’s modulus, and is the Poison ratio of the plate. The
analytic solution is 0.016475 m, and the result of the numerical analysis is
converged to the analytic solution when the number of element increases. This
shows the flexible dynamics code for the plate is reliable.
72
3.2. Verification of a simulation program with
OrcaFlexTM
The verification of the code of elements has shown in preceding chapter.
This chapter shows the verification of the code with multibody example by
comparing the result of commercial code OrcaFlexTM.
3.2.1. Modeling of the verification example
Fig. 28 Configurations of the modeling of the verification example
To verify the flexible multibody dynamics code, some more coplex model
has compared with commercial code OrcaFlexTM. Fig. 28 shows configurations
of the modeling of the example. In this example, a floating barge is connected
with a riser which is modeled as an ANCF beam. The platform and the riser
and riser and seabed are connected by the ball joints. The barge is connected
with a spring that oscillates the barge dynamically. Table 5 and Table 6 show
the properties of the barge and the riser used in the model.
73
Table 5 Properties of the barge used in the verification example
Barge properties Value
Length 110 m
Breadth 46 m
Depth 7.5 m
Draft 3.75 m
Weight 19,445.375 ton
Center of gravity (55, 23, 3.75) m
Table 6 Properties of the riser used in the verification example
Riser properties Value
Length 300 m
Outer diameter 0.6 m
Inner diameter 0.4 m
Material density 7,850 kg/m3
Young’s modulus 210 GPa
Poison ratio 0.3
2nd moment of area 0.010529 m4
Sea water density 1,025 kg/m3
Weight in water 1,056,438
The dimensions of the barge are (110 m, 46 m, 7.5 m, 3.75 m) for (L, B, D,
T) and its center of gravity is located on (55 m, 23 m, 3.75 m). The length of
the riser is 300 m with 0.6 m and 0.4 m for outer diameter and the inner diameter,
and the material of the riser is mild steel. The weight in water is calculated by
subtraction of the displacement weight of the sea water from the light weight
of the riser as
2 2 2
4 4riser SW
OD ID ODWeight in water gL
, (182)
wherein riser and SW are the density of the matrial of the riser, and sea
74
water, g is the acceleration of the gravity which is 9.81 m/s, L is the length
of the riser, and OD and ID are the outer diameter and the inner diameter.
The second moment of the riser is obtained from
4 42 / 64nd moment of inertia OD ID . (183)
3.2.2. Simulation results of the verification example
Fig. 29 Screen shot of the simulation by in-house code and OrcaFlexTM
Fig. 29 shows screen shot of the simulation by in-house code and
OrcaFlexTM, and Fig. 30 and Fig. 31 shows the comparison of the heave motion
of the platform with one element riser and five element riser.
Fig. 30 Comparison of the heave motion of the platform with 1 element riser
75
Fig. 31 Comparison of the heave motion of the platform with 5 element riser
The trend of the heave motion of the in-house code follows the result of
OrcaFlexTM. However, the perturbation is shown in the result of the in-house
code, because the simulation of the OrcaFlex starts in static equilibrium
position, but in-house code could not calculate the static equilibrium position.
The reason of the difference of height of the heave motion is assumed that the
difference of the boundary condition where the boundary condition of the riser
in OrcaFlex is free in rotational degree of freedom, but the boundary condtion
of the riser in the in-house code is restrained due to the limitation of the
implementation of the joint constraint.
76
77
78
79
80
81
Fig. 32 Comparison of the behavior of one element riser
82
83
84
85
86
Fig. 33 Comparison of the behavior of five elements riser
87
Fig. 32 and Fig. 33 show the behavior of the one element riser and the five
element riser in time domain for zero to one hundred seconds. The top end of
the graph is a ball joint that connect the barge and riser, and the bottom end of
the graph is a fixed joint that conncet the riser and seabed. When the simulation
start, the spring pulls the barge to the negative X-axis direction. The riser
deforms due to the motion of the barge, Therefore, when the barge reaches near
-10 m. The riser pulls the barge to the positive X-axis. Therefore, the barge
moves to the positive X-axis direction after 40 seconds. The deformed riser also
pull the barge to the downward. Therefore, the barge slightly has submerged
during the simulation. The results show that the behavior of the five element
riser is closer to the behavior of riser in OrcaFlexTM.
.
88
3.3. Modeling of the drilling rig with a riser
This chapter describes the modeling of the drilling rig with a riser.
3.3.1. Configurations of the drilling rig with a riser
Fig. 34 Configurations of the drilling rig with a riser
An offshore drilling rig is a vessel for drilling operation to develop or
explorer offshore oil and gas reservoirs. An offshore drilling rig while operation
has complex system that consists of drilling rig platform, upper flex joint,
telescopic joint, riser, lower flex joint, BOP (Blow Out Preventer), and riser
tensioner as Fig. 34. This system helps stable drilling operation in ocean
environment. A riser is a pipe which is the passage of a drill string and mud that
89
carries out rock cuttings. An upper flex joint is a mechanical joint that
interconnects a platform and a telescopic joint. The upper flex joint makes the
riser rotate freely. A telescopic joint is a mechanical joint that interconnects an
upper flex joint and a riser. Telescopic joint can rotate freely on the axix of the
joint, and it allows the translational motion following the axis as a telescope
named itself. The telescopic joint allows the longitudinal motion of the riser
due to the motion of the platform. A lower flex joint interconnects a riser and a
BOP. Kinematic characteristic of a lower flex joint is same as upper flex joint.
BOP is the equipment that prevents blow out of the well. A BOP is fixed on a
well head and connected with a lower flex joint. Wireline riser tensioner is an
equipment that pull the riser with constant tension to prevent buckling and
heave motion of the riser.
3.3.2. Wireline riser tensioner
Fig. 35 Model of pressure system for riser tensioner
90
Wireline riser tensioner is an equipment that provides constant tension to
the riser by a pneumatic system. The riser tensioner system is a kind of massive
spring that consists of several pneumatic cylinders. Thus, the riser tensioner
system can keep the force applied to the riser string nearly constant and
minimize the movement of the riser string, independent of the movement of the
platform due to waves (Lee et al., 2015).
Fig. 36 Configurations of a wireline riser tensioner and a riser
The pneumatic system in the wireline tensioner which absorbs forces and
the motion delivered from the platform consists of pistons & upper sheaves,
accumulator, and APV (Air Pressure Vessel) as Fig. 35. One side of the pistons
& upper sheaves is connected with the accumulator which is hydraulic-
pnuematic cylinder, and the other side is connected with a riser by wirelines as
91
Fig. 36. The APV is filled with pressurized nitrogen gas, and this absorbs forces
and motions through compression and expansion of the gas. Due to this reason,
wireline riser tensioner system can be regarded as dynamic non-linear spring-
mass-damper system. The elastic coefficient of the wireline riser tensioner is
derived from the equation of state, and elastic coefficient can be written as
,0
1,0
11
1 ( ) /p gas
p p p gas
A Pk
y y A y y V
, (184)
Wherein pA is the area of the cylinder, y is the z-position of the cylinder,
py is the z-position of the deck on the platform, ,0gasP is the initial pressure
of the APV, and ,0gasV is the volume of the APV.
The friction of the cylinder is calculated by Stribeck friction model. Stribeck
friction is occurred when the fluid or the lubricant oil contacts to the surface of
the equipment or cylinder. Stribeck friction decreaces when the speed of the
cylinder increases under the Stribeck velocity. Stribeck friction is sum of the
friction which terms of the velocity ,fr vF , and the pressure ,fr pF as
, exp
piston
fr v c s c v pistonl
vF F F F k v
v
, maxfr p p sF k p F
(185)
wherein lv is Stribeck velocity, pistonv is velocity of the piston, sF is static friction, and
cF is Coulomb friction.
92
3.3.3. Modeling information of the drilling rig with a riser
Fig. 37 Bodies and joints information of a drilling rig with a riser
Information of the bodies and the joints of the main example is shown in
Fig. 37. There are 7 rigid bodies which are piston & upper sheave 1 and 2,
platform, telescopic joint upper and lower, lower flex joint upper, and BOP,
and one flexible body, riser, in the main example. The hydrostatic and dynamic
forces are applied to the platform, and the current force is appled to the riser.
Table 7 sumarizes the body information of the main example.
Table 7 Body information of the main example
Body
number Body type Name
Body 1
Rigid
Piston & upper sheave 1
Body 2 Piston & upper sheave 2
Body 3 Platform
Body 4 Telescopic joint (upper)
Body 5 Telescopic joint (lower)
Body 6 Flexible Riser
Body 7 Rigid
Lower flex joint (upper)
Body 8 Lower flex joint (lower) & BOP
93
The bodies are interconnected with joints as Fig. 37. Piston & upper sheave
1 and 2 are connected to platform by a cylindrical joint, platform and telescopic
joint (upper) are interconnected by a ball joint, telescopic joint (upper) and
telescopic joint (lower) are interconnected by a cylindrical joint, telescopic joint
(lower) and riser are interconnected by a fixed joint, riser and lower flex joint
(upper) are interconnected by a fixed joint, lower flex joint (upper) and lower
flex joint (lower) & BOP are interconnected by a balljoint, and BOP is fixed at
the seabed. Table 8 sumarizes the joint information of the main example.
Table 8 Joint information of the main example
Joint
number Joint type Connection information
Joint 1 Cylindrical
Piston & upper sheave 1 - Platform
Joint 2 Piston & upper sheave 2 - Platform
Joint 3 Ball Platform – Telescopic joint (upper)
Joint 4 Cylindrical Telescopic joint (upper) – Telescopic joint
(lower)
Joint 5 Fixed Telescopic joint (lower) – Riser
Joint 6 Fixed Riser – Lower flex joint (upper)
Joint 7 Ball Lower flex joint (upper) – Lower flex joint
(lower) & BOP
Joint 8 Fixed Lower flex joint (lower) & BOP - seabed
The properties of the platform and the riser are described in Table 9 and
Table 10. The main dimensions of the platform are (108 m, 77.57 m, 41.4 m,
23.0 m) for (L, B, D, T), and the weight of the platform is 54,530 ton. The
length of the riser is 300 m, with 0.7 m for outer diameter and 0.4 m for inner
diameter. The steel properties are used for the riser material properties. The
wave height and the period for environmental conditions of the example are 3
m and 10.5 sec with or without 3 m/s current.
94
Table 9 Properties of the semi-submersible platform used in the main example
Platform properties Value
Length 108 m
Breadth 77.57 m
Height 41.4 m
Draft 23.0 m
Weight 54,530 ton
Table 10 Properties of the riser used in the main example
Riser properties Value
Length 300 m
Outer diameter 0.7 m
Inner diameter 0.4 m
Material density 7,850 kg/m3
Young’s modulus 210 GPa
Poison ratio 0.3
Drag coefficient 0.65
Inertia coefficient 1.6
Wave height 3 m
Wave period 10 sec
Underwater current velocity 3 m/s
Number of element 10
95
3.4. Numerical simulation results of a drilling rig
with a riser
Fig. 38 Screen shot of the drilling rig simulation
This chapter shows the simulation result of the motion analysis of the
drilling rig with the riser. The properties of the bodies are described in
preceding chapter. Fig. 38 is the screen shot of the simulation program. The
hydrodynamic force is applied after 5 seconds has passed with ramp period 30
seconds.
3.4.1. 10 elements without current
Fig. 39, Fig. 40, and Fig. 41 show the motion of the platform without current
under 3 m wave height, and 10.5 sec wave period.
Fig. 39 Surge motion of the platform without current
96
Fig. 40 Heave motion of the platform without current
Fig. 41 Roll motion of the platform without current
From the result of the Fig. 39, platform shows short period motion due to
the hydrodynamic force, and long period motion due to the stiffness of the riser.
All graphs shows the oscillation begin after 5 seconds because the
hydrodynamic force is applied after 5 seconds with 30 seconds ramp period.
The behavior of the riser during the simulation time is as follows.
97
98
99
100
101
Fig. 42 Behavior of the riser without current
Fig. 42 shows the behavior of the riser without the current through out the
simulation time. The one check that the riser deformes due to the behavior of
the platform. Ther riser deforms after 5 second because the hydrodynamic force
is applied to the platform after 5 second. The current force is not take account
into this result, the hydrodynamic force makes the motion of the patform, and
it makes the riser deform. By the response of the platform due to the wave, the
platform moves between -1.5 m to 1.5m, and the maximum deformation of the
riser is in this range. From Fig. 42, riser at 50 m 310 m deforms frequently, by
means that parts of the riser has to be cautious of the fatigue.
102
3.4.2. 10 element with 3 m/s current
Fig. 43, Fig. 44, and Fig. 45 show the motion of the platform with 3 m/s
current under 3 m wave height, and 10.5 sec wave period.
Fig. 43 Surge motion of the platform with 3 m/s current
Fig. 44 Heave motion of the platform with 3 m/s current
Fig. 45 Roll motion of the platform with 3 m/s current
Not like Fig. 42, Fig. 43 shows only short period motion due to the
hydrodynamic force because the current force pull the riser continuously.
103
Therefore, there is no long period motion due to the riser. The platform is
pushed to 1 m X-axis by the current froce, and shows surge motion due to the
hydrodynamic force. The hydrodynamic force is also applied after 5 second.
However, the current force is applied with the beginning of the simulation.
Therefore the riser deforms immediately when the simulation begin as Fig. 46.
The heave and roll motion of the platform is almost same as the model without
current, because the hydrodynamic properties are not changed.
104
105
106
107
Fig. 46 Behavior of the riser with 3m/s current
108
Fig. 46 shows the behavior of the riser with 3 m/s current through out the
simulation time. Therefore the riser deformes due to the current immediately.
General shape of the riser in Fig. 46 look like a bow, and the maximum
horizontal deformation is about 11 m to X axis. The major response of the riser
is dominated by the current. Comparing with the nocurrent condition, the
deformation due to the current is larger than the deformation due to the
hydrodynamic force. Therefore, the current is the crucial point for the riser
behavior and its design. In this model, the current profile is appled as uniform
distribution. Therefore, the maximum deformation of the riser is shown at the
middle of the riser. The designer have to check the max bending moment at the
point. About 65 m of the riser is the point of inflection which deforms
frequently. This point has to be cautious about fatigue damage.
109
Conclusions and future works
4.1. Conclusions
In this paper, the coupled analysis of the drilling rig and the riser using
flexible multibody dynamics has performed.
At first, we formulated equations of the motion of the beam element and the
plate element with absolute nodal coordinate formulation, and verified the
element with the analytic solution of the pure bending of the cantilever beam,
and the simple supported plate.
For the next, joint constraint equations for the flexible body and the rigid
body has derived. The developed code has verified with commercial code in
multibody scale by means that the simulation example consists several bodies
and joint with several external forces.
Finally, the coupled analysis of the drilling rig and the riser has performed.
This main example consists of 8 bodies and 8 joints, and the hydrostatic force,
hydrodynamic force, and current force has been applied to the model. With this
example, we can predict response of platform-riser coupled system in ocean
environment, and dynamic loads on the riser, joints and tensioner lines in time
domain. The result can be used for fatigue and stress analysis of the riser.
The developed flexible multibody dynamics code is not only for the coupled
analysis of drilling rig and the riser, but also can be used for various engineering
field that requires dynamic analysis of the mechanical system that including
deformable bodies.
4.2. Future works
110
The developed flexible multibody dynamics code can not find static
equilibrium position. Therefore, additional damping forces are applied to find
static equilibrium position at the beginning stage. This leads the computational
time increasing. Therefore, static equilibrium calculation method has to be
developed to take less efforts, and increasing the accuracy of the dynamic
analysis.
The developed code uses 4th order Runge-Kutta method for time integration.
However, 4th order Runge-Kutta is an exiplicit numerical integration method.
Explicit method is unstable for high stiffness problem, and it requires small
time step which makes the code requires many computational efforts. To
overcome this limitation, implicit numerical integration module has to be
developed.
Another limitation of the code is simulation speed is slow due to the size of
the matrix of the equations of the motion is huge. Therefore, another form of
the equations of the motion has to be formulated such as using polar
decomposition of the mass matrix.
For the last, various joint constraint equations for interconnecting flexible
bodies and rigid rigied bodies has to be developed. This paper derived only
fixed joint for flexible-rigid connection which is fixed or clamped boundary
condition in FEM analysis. Therefore, other flexible joints have to be developed.
111
References
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Ocean Engineering, 32, pp.802-816.
[2] Low, Y.M., Langley, R.S., 2008, “A hybrid time/frequency domain approach
for efficient coupled analysis of vessel/mooring/riser dynamics”, Ocean
Engineering, 35, pp.433-446.
[3] Santillan, S.T., Virgin, L.N., Plaut, R.H., 2010, “Static and Dynamic
Behavior of Highly Deformed Risers and Pipelines”, Journal of Offshore
Mechanics and Arcting Engineering, 132, 021401-1.
[4] Yang, M., Teng, B., Ning, D., Shi, Z., 2012, “Coupled dynamic analysis for
wave interaction with a truss spar and its mooring line/riser system in time
domain”, Ocean Engineering, 39, pp.72-87.
[5] Park, K.P., 2011, Flexible mutibody dynamics of floating offshore wind
'turbine in marine operation (Doctorial dissertation), Seoul National
University.
[6] Bauchau, O.A., 2011, Flexible Multibody Dynamics, Springer, Newyork.
[7] Shabana, A.A., 2005, Dynamics of multibody systems, Cambridge
University Press, New York.
[8] Berzeri, M., Shabana, A.A., 2000, “Development of Simple Models for
theElatic Forces in the Absolute Nodal Coordinate Formulation”, Journal
of Sound and Vibration, 235(4), pp.239-565.
[9] Dmitrochenko, O.N., Pogorelov, D.Yu, 2003, “Generalization for plate
112
finite elements for Absolute Nodal coordinate Formulation”, Multibody
System Dynamics, 10, pp.14~43.
[10] Yoo, W.S., Lee, J.W., Park, S.J., Sohn, J.H., Pogorelov, and Dmitrochenko,
2004, “Large Deflection Analysis of a Thin Plate : Computer Simulation
and Experiments”, Multibody System Dynamics, 11, pp.185~208.
[11] Nikravesh, P.E., 1988, Computer Aided Analysis of Mechanical system,
Prentice Hall, Englewood Cliff.
[12] Sugiyama, H., Escalona, J.L., Shabana, A.A., 2003, “Formulation of three-
dimensional Joint Constraints Using the Absolute Nodal Coordinates”,
Nonlinear Dynamics, 31, pp.167-195.
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[14] Cummins, W.E., 1962., The impulse response function and ship motions.
Schiffstechnik 9, 101-109.
[15] Det Norske Veritas, 2002. WADAM (Wave Analysis by Diffraction and
Morison) theory. SESAM’s user manual.
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edition, PWS Publishing.
[17] Timoshenko, S., and Woinowsky-Kriger, 1959, Theory of Plates and
Shells, McGraw-Hill, New York.
[18] Lee, H.W., Roh, M.I., Ham. S.H., and Ha, S., 2015, “Dynamic simulation
of the wireline rier tensioner system for a mobile offshore drilling unit
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495.
113
Appendix
Derivation of the curvature in small longitudinal
deformation
Recall that equation (36),
2
3 2
d
dx
r r rκ r
r
In case of the small deformation, the curvature of the beam can be represent
as the magnitude of the second derivative of the position vector. In this chapter
shows the derivation of the curvature with small deformation as above.
Appendix-Fig 1 Derivation of the curvature with small deformation
Appendix-Fig 1 shows the geometric features of the infinitesmall curvature.
The definition of the curavature κ is the inverse number of the radius of the
curvature as
1
κ .
From the Appendix-Fig 1,
114
d
tand
y
x .
Derivative of the tan with respect to the x becomes
2
2 22
d d d(sec ) 1 tan
d d d
x
x x y
.
Therefore,
2
2
2
dd
ddd
1d
yx
xyx
.
Then, the infinitesimal arclenth ds is obtained from Pythagorean theorem,
2
2 2 dd d d d d 1
d
ys x y x
x
.
And then, the curvature is obtained as
2
2
32 2
dd1 d d
dd
1 dd
yx
xs
yx
x
.
If the deformation is small, high order term can be neglected. Then the curvature
becomes
2
22
3 22 2
dd1 dd
dd
1 dd
yx yx
xy
xx
.
Thus, equation (36) has been derived
115
Explicit matrix expression for plate element
Basic notation
Differenciation of shape funtions of the plate with respect to parameters 1p
and 2p is denoted by upperindices:
iklkl
i
SS
p
,
2ijklkl
i j
SS
p p
.
Their integration over the plate surface is denoted by double bar:
1 2
0 0
d da b
kl klS p p S .
A bar and hats (one or two) mark integration of one-dimensional shape
functions:
11 10
ˆ ˆˆ da i j
ij k lkl i j
s sS p
p p
, 22 2 2 20
ˆ ˆˆ ˆ ˆ ˆ ˆˆ db i j i j
ijij k l m nklmn i j i j
s s s sS p
p p p p
Notation for integration over the plate element surface is as follows
1 2
0 0
d d da b
P
P p p .
Expressions for Mid-Plane Forces
Six index symbols
116
1d
2i j i j j iklrs kl rs kl rsP
S S S S P S
1 1 1 1
1 1
11 001 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
klrs kl rs k l r sP P
a b
k r l s kr ls
S S P s s s s Pp p
s s p s s p S S
S
2 2 2 2 00 111 2
2 2 0 0
ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd d d da b
klrs kl rs k l r s k r l s kr lsP PS S P s s s s P s s p s s p S S
p p
S
1 2 2 1 1 2 2 1
1 1
1 2 1 20 0 0 0
10 10 10 10
1d
2
1 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d2 2 2
1 ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd d d d2
ˆ ˆ1 ˆ ˆ ˆ ˆ2
klrs klrs kl rs kl rsP
k l r s k l r sP
a b a b
k r l s k r l s
kr sl rk ls
S S S S P
s s s s s s s s Pp p p p
s s p s s p s s p s s p
S S S S
S S
Ten index symbols
;;
1d
2i j i j i j i jklrs mnqp klrs mnpqP
S S P S
1 1;1 1 1 1 1 1 1 1 1 1;
1 1 1 1
1111 00001 20 0
d d
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d
ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd d
klrs mnpq klrs mnpq kl rs mn pqP P
k l r s m n p qP
a b
k r m p l s n q krmp lsbq
S S P S S S S P
s s s s s s s s Pp p p p
s s s s p s s s s p S S
S
117
2 2;2 2 2 2 2 2 2 2 2 2;
2 2 2 2
0000 11111 20 0
d d
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d
ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd d
klrs mnpq klrs mnpq kl rs mn pqP P
k l r s m n p qP
a b
k r m p l s n q krmp lsbq
S S P S S S S P
s s s s s s s s Pp p p p
s s s s p s s s s p S S
S
1 2;1 2 1 2 2 1 1 2 2 1;
1100 1100 1100 1100 1100 1100 1100 1100
1d
4ˆ ˆ ˆ ˆ1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
4
klrs mnpq kl rs kl rs mn pq mn pqP
kmpr qsnl rmkp lqns kpmr nslq prkm nlqs
S S S S S S S S P
S S S S S S S S
S
1 1;2 2;
1 1 2 2
1100 11001 2
0 0
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ d
ˆˆ ˆˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆd d
klrs mnpq k l r s m n p q
P
a b
k r m p n q l s krmp nqls
s s s s s s s s Pp p p p
s s s s p s s s s p S S
S
2 2;1 1 1 1;2 2 1100 1100; ;
ˆˆ ˆklrs mnpq mnpq klrs mpkr lsnqS S S S
Expressions for Transverse Forces
Six index symbols
dijij ij ijmnpq mn pqP
S S P S
2 2
1111 11 112 21 1
22 001 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
mnpq mn pq m n p qP P
a b
m p n q mp nq
S S P s s s s Pp p
s s p s s p S S
S
118
2 2
2222 22 222 22 2
00 221 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
mnpq mn pq m n p qP P
a b
m p n q mp nq
S S P s s s s Pp p
s s p s s p S S
S
2 2
1122 11 222 21 2
20 201 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
mnpq mn pq m n p qP P
a b
m p q n mp qn
S S P s s s s Pp p
s s p s s p S S
S
2 2
2211 1122 22 112 22 1
20 201 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
mnpq pqmn mn pq m n p qP P
a b
p m n q pm nq
S S P s s s s Pp p
s s p s s p S S
S S
2 2
1212 12 12
1 2 1 2
11 111 2
0 0
ˆ ˆˆ ˆ ˆ ˆd d
ˆˆ ˆˆ ˆˆ ˆ ˆ ˆd d
mnpq mn pq m n p qP P
a b
m p n q mp nq
S S P s s s s Pp p p p
s s p s s p S S
S
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국문 초록
유연 다물체계 동역학을 이용한 해양 시추
리그의 라이저 거동 해석
조선 해양 분야에서 운동 해석은 조선 생산 시 블록 인양
작업 및, 해양 설치 작업 시의 동적 하중 결정, 해양 시추
리그의 시추 작업 및 해양플랜트 생산 작업 중의 라이저
(riser)와 플랫폼(platform)의 연성 운동 해석 등 다양한 분야
에 적용되고 있다. 이 때, 물체를 강체로 가정할 경우 물체
의 변형이 전체 시스템에 미치는 거동을 반영 할 수 없고,
이는 해양 시추 리그와 라이저의 연성 운동과 같이 실제
물체의 대변형(large deformation)이 전체 시스템의 거동에
영향을 미치는 경우 해석이 불가능 하다. 따라서 운동 해
석 시에 물체의 변형을 고려하여 운동 해석을 수행하여야
한다.
본 연구에서는 물체의 변형을 고려할 수 있는 다물체계
(multibody system)의 운동을 기술하기 위한 운동방정식을
구성하는 방법에 대한 연구를 수행하였다. 유연체의 변형
을 고려하기 위하여 ANCF(Absolute Nodal Coordinate
Formulation)를 이용하여 3차원 보(beam)의 운동방정식을 유
도하였고, ANCF 보와 강체와의 구속조건을 유도하여 이를
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해양 시추 리그의 라이저 거동 해석에 적용하였다.
해양 시추 리그의 라이저 거동 해석을 위하여 본 연구에서
는 해양 시추 리그, 해양 시추 리그와 라이저를 연결하는
텔레스코픽 조인트(telescopic joint), 라이저와 BOP(Blow Out
Preventer)를 연결하는 로워플렉스 조인트(lower flex joint),
해저면에서 폭발을 방지하는 BOP를 강체로 모델링하고 라
이저를 ANCF 보로 모델링 하였다. 또한 라이저의 인장력
(tension)을 유지시켜 라이저의 좌굴(Buckling)을 방지하고,
리그의 상하동요 운동에도 라이저의 위치를 유지 시켜주는
라이저 인장장치(riser tensioner)를 공압 스프링(pneumatic
spring)으로 모델링하였다. 각각의 물체들은 구속조건을 고
려하여 고정 관절(fixed joint), 실린더 관절(cylindrical joint),
구형 관절(ball joint)을 이용하여 연결하였고 작용하는 외력
으로 시추 리그에 작용하는 유체 동역학적 힘과 정역학적
힘을, 라이저에 작용하는 조류력을 고려하여 해양 시추 리
그와 라이저의 연성 해석(coupled analysis)을 다물체계 동역
학을 운동방정식을 구성하여 수행하였다.
본 연구를 요약하면 1) ANCF 방식으로 3차원 보의 운동방
정식을 구성하여 운동 해석 시 보의 변형을 고려하였고, 2)
강체와 ANCF 보의 구속조건을 유도하여 강체와, ANCF 보
의 연성해석을 수행하였으며, 3) ANCF 보에 작용하는 다양
한 외력을 고려하였다. 이를 바탕으로 본 연구에서는 4) 보
의 유연성을 고려할 수 있는 범용 동역학 해석 코드를 개
발하고 5) 최종적으로 해양 시추 리그와 라이저의 거동을
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연성 하여 해석하고, 이를 상용코드와 비교하여 검증하였
다.
Keywords: 유연 다물체계 동역학, 다물체계 동역학, 라이저, 해양
시추 리그, 동적 거동, 절대 절점 좌표계법(ANCF)
Student number: 2014-20650