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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

Qq

. (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

[1] Garrett, D.L., 2005, “Coupled analysis of floating production systems”,

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.

[13] Cha, J.H., Roh, M.I. and Lee, K.Y., 2010. “Integrated simulation

framework for the process planning of ships and offshore structures.”

Robotics and Computer-Integrated Manufacturing Journal, 26(5), 430-453.

[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.

[16] Gere, J.M. and Timoshenko, S.P., 1997. Mechanics of materials, 4th

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

based on multibody system dynamic”, Ocean Engineering, 106, pp.485-

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

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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