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Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0314-2
Derivation of equations of motion of an unwinding cable
from a cylindrical spool package† Jae-Wook Lee1, Deuk-Man An2 and Wan-Suk Yoo2,*
1Research Institute of Mechanical Technology, Pusan National University, Busan, 609-735, Korea 2School of Mechanical Engineering, Pusan National University, Busan, 609-735, Korea
(Manuscript Received October 4, 2010; Revised February 7, 2011; Accepted February 7, 2011)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract In this study, the transient analysis of a cable unwinding from a cylindrical spool package is studied. Cables are withdrawn with a con-
stant velocity through a fixed point which is located along the axis of the package. When the cable is flown out of the package, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force, and fluid-resistance force are acting on the cable. Consequently, the cable becomes to undergo very nonlinear and complex unwinding behavior which is called unwinding balloon. In this paper, to prevent the problems during unwinding such as tangling or cutting, unwinding behaviors of cables in transient state were de-rived and analyzed. First of all, the governing equations of motion of cables unwinding from a cylindrical spool package were systemati-cally derived using the extended Hamilton’s principles of an open system in which mass is transported at each boundary. And the modi-fied finite difference methods are suggested to solve the derived nonlinear partial differential equations. Time responses of unwinding cables are calculated using Newmark time integration methods.
Keywords: Behavior of unwinding cable; Unwinding balloon; Extended Hamilton’s principle; Modified finite difference methods ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction
Unwinding of a thread had been widely applied in the field of textile engineering [1, 2], in which their results were con-fined to steady states. And in these days, these are also applied in unwinding techniques of long wires are attached in guided weapons [3, 4]. The length of wires in guided weapons ex-ceeds tens of kilometers, which are wound in a cylindrical spool package. When the wire is about to fly out of the pack-age, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force and fluid-resistance forces are acting on the wire. Thus, the wire undergoes very nonlinear and complex unwinding behaviors which are called as unwinding balloon.
To analyze and avoid unwinding difficulties such as tan-gling, the complex unwinding behaviors should be verified. Since the balloon shapes during unwinding highly depend on the tension, the effect of tension should be considered.
In this paper, to analyze unwinding problems of a cable such as tangling, the unwinding behaviors of cables from cy-lindrical spool packages are studied for a transient state. First,
the governing equations of motion of cables unwinding from a cylindrical spool package are systematically derived using the extended Hamilton’s principle [5] proposed for an open sys-tem, in which mass can be transported at each boundary. The derivation is explained in section 2, and a modified finite dif-ference methods is suggested in section 3 to numerically solve the nonlinear partial differential equations derived in section 2. Time responses of unwinding cables were calculated using the Newmark time integration methods [6, 7]. In section 4, tran-sient unwinding behaviors of cables from the cylindrical spool package are shown according to magnitudes in tensile forces. Finally, in section 5, conclusions of this paper are presented and the future work is proposed. 2. Governing equations of unwinding motion 2.1 Extended Hamilton’s principle for unwinding
Winding can be divided into outer and inner winding as shown in Fig. 1, and the unwinding phenomenon from the outer winding can be systematically drawn as Fig. 2. When a cable is withdrawn with a constant velocityV through the point O , the cable flies off the package from one point L . The points O and L are gener-ally called guide eyelet and lift-off point, respectively. The verti-cal distance between the guide eyelet and the lift-off point is called the balloon height, which is shown as BH in Fig. 2. The
† This paper was recommended for publication in revised form by Editor Yeon June Kang
*Corresponding author. Tel.: +82 51 510 1457, Fax.: +82 51 581 8514 E-mail address: [email protected]
© KSME & Springer 2011
1288 J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
region from the guide eyelet to lift-off point is called as balloon region, and the radius of the spool packages is rC .
The system with an unwinding cable is a time varying system in which the total mass of the system is changing. So, it is difficult to directly apply Hamilton’s principle to the unwinding cable which is a time varying and open system. In this paper, an ex-tended Hamilton principle proposed by McIver [5] for an axially moving plate was extended for the unwinding analysis through modifications.
The principle of virtual work for a closed system is in Eq. (1).
( ) ( )( )0
cp p p cc R t
DL W v R dVDt
δ δ ρ δ+ − ⋅ =∫∫∫ , (1)
where ( )cR t is the closed system which can move freely and bounded by the surface ( )cB t and , p pR v are position and velocity vectors of a particle, respectively. pρ is the density of a particle. The Lagrangian ( ) ( )c c
L K E= − is composed of the kinetic energy ( )c
K and potential energy ( )cE of the
conservative system. And Wδ is the virtual work performed
by the generalized forces undergoing generalized virtual dis-placements and /D Dt is the material derivative following a particle or a specific collection of particles.
The Hamilton principle of Eq. (2) is obtained by integrating Eq. (1) with respect time between two instants 1t and 2t .
( )2 2
1 1
0t t
ct tL dt Wdtδ δ+ =∫ ∫ . (2)
To extend the Hamilton Principle of a closed system to a
system of changing mass, McIver [5] introduced the time varying open system such as Fig. 3, and derived the extended Hamilton principle of a time varying open system using the general transport theorem, is written in Eq. (3).
( )( )
( )( )
( )( )( )ˆ
o c
o
o cR t R t
B pB t
d DdV dVdt Dt
V v ndA
=
+ − ⋅
∫∫∫ ∫∫∫
∫∫, (3)
where BV , pv is the velocity of moving boundary and the velocity of the particle which is getting out through the open boundary, respectively, and n is the outward normal vector at the open boundary.
Applying the general transport theorem, Eq. (3) into the vir-tual work equation, Eq. 1 can be written as Eq. (4).
( ) ( )( )( )
( )( ) 0
ˆ
0.
o
o
p p B po B t
p p pR t
L W v R V v ndA
d v R dVdt
δ δ ρ δ
ρ δ
+ + ⋅ − ⋅
− ⋅ =
∫∫
∫∫∫ (4)
Integrating with respect to time as before between 1t and
2t when the system configuration is prescribed, the extended form of Hamilton’s principle for a system of changing mass becomes Eqs. (5) and (6).
( )2 2
1 1
0t t
ot tL dt Hdtδ δ+ =∫ ∫ , (5)
( )( )( )ˆ
op p p B pB t
H W v R V v ndAδ δ ρ δ= + ⋅ − ⋅∫∫ , (6) where ( ) ( )o o
L K E= − is the Lagrangian of the system con-tained within the open control volume, ( )oR t and Wδ is the variation of the virtual work performed on this same sys-tem. The last integral term on the right side of Eq. (6) may be thought of as a virtual momentum transport across the open surface ( )oB t . 2.2 Energies on unwinding cables
Assume the cable is unwounding through a guide eyelet O with a contant unwinding velocity of V . The guide eyelet is located along the axis of the package, and the vertical distance between the guide eyelet O and the lift-off point L is the balloon height BH . The position and velocity vector of one element which is on unwinding cables at arbitrary time can be expressed as Eqs. (7) and (8), respectively.
(a) Outer winding packages (b) Inner winding packages Fig. 1. Payout dispenser of cylindrical spool packages [8] (AutoFlugGMBH).
ω
ˆreeθ
ˆze
O′rC
BH
P
R
V
O
L
Fig. 2. Unwinding behaviors of cables from cylindrical spool package.
Fig. 3. Open and closed system for the extended Hamilton’s principle [3].
J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296 1289
ˆ ˆr zR re ze= + , (7)
v DR Rω= + × , (8)
where , , r zθ are cylindrical coordinates corresponding to the basis vector, ˆ ˆ ˆ, , r ze e eθ , respectively, and D is the dif-ferential operator like Eq. (9). Due to constant unwinding velocity, the cable has the rotational unwinding velocity ω about the cylindrical package.
sD Vt t s t s∂ ∂ ∂ ∂ ∂
= + = −∂ ∂ ∂ ∂ ∂
. (9)
Thus, the kinetic energy and the potential energy which are
occurring during unwinding of cables can be expressed as Eqs. (10) and (11), respectively.
( )0
12
LK A v v dsρ= ⋅∫ (10)
( )( )0
12
L
s sE T s R R ds= ⋅∫ (11)
where , Aρ is a density and cross sectional area of a cable, respectively, and T is a tension of a point P. And ( ) ( ) /t
t≡ ∂ ∂ and ( ) ( ) /ss≡ ∂ ∂ can be expressed, re-
spectively. The variation of Eqs. (10) and (11) are the same as Eqs. (12)
and (13), respectively.
( )0
LK A v v dsδ ρ δ= ⋅∫ (12)
( )( )0
L
s sE T s R R dsδ δ= ⋅∫ . (13)
The virtual work by fluid-resistance forces can be repre-sented as Eq. (14) and the virtual momentum transport across the open boundary can be expressed as Eq. (15).
( )NC 0,
L
aW F s t Rdsδ δ= ⋅∫ , (14)
( )( ) ( )( ) ( )( )
MT 0 0
0 .
l l
l
s s
A
s
W v R V dA AV v R
AV DR R R
δ ρ δ ρ δ
ρ ω δ
= ⋅ − = − ⋅
= − + × ⋅
∫ (15)
2.3 Derivation of governing equations of motion
The governing equations of motion can be derived as Eq. (16) using the variational forms of each energy written in Eqs. (11)-(14) and the extended Hamilton’s principle of a open system of Eq. (5).
( ) ( )( )( )
( ) ( )( )
2 2
1 1
2
1
2
1
0 0
0
0
,
0l
t L t L
s st t
t L
at
st
t
A v v dsdt T s R R dsdt
F x t Rdsdt
AV DR R R dt
ρ δ δ
δ
ρ ω δ
⋅ − ⋅
+ ⋅
⎛ ⎞+ + × ⋅ =⎜ ⎟⎝ ⎠
∫ ∫ ∫ ∫
∫ ∫
∫
(16)
Because the first term of Eq. (16) is the kinetic energy, rear-
ranging it using the velocity equation, it can be expressed as Eq. (17).
( )
( )( )( )( )( )( )
( )( )
2
1
2
1
2
1
2
1
2
1
0
0
0
2
0
2
0
=
t L
t
L t
t s t
L t
tt ts tt
t L
t st
t L
ts ss st
A v v dsdt
A R VR R R ds
A R VR R Rdtds
A VR V R V R R dt
A VR V R V R Rdsdt
A R
ρ δ
ρ ω δ
ρ ω δ
ρ ω δ
ρ ω δ
ρ ω
⋅
− + ×
− − + ×
− − + ×
+ − + ×
+ − ×
∫ ∫
∫
∫ ∫
∫
∫ ∫( ) ( )( )( )2
1 0
t L
t stVR R Rdsdtω ω δ− − × ×∫ ∫
(17)
If differential operators are defined as Eq. (18), Eq. (17) can
be rewritten as Eq. (19).
2 2, 2t s tt ts ssDR R VR D R R VR V R= − = − + . (18)
( )
( ) ( )( )( )( )( )( )
2
1
2
1
2
1
2
1
0
2
0
0
0
= 2
t L
t
L t
t
L t
t s t
t L
t st
A v v dsdt
A D R DR R Rdtds
A R VR R R ds
AV R VR R R dt
ρ δ
ρ ω ω ω δ
ρ ω δ
ρ ω δ
⋅
⎡ ⎤− + × + × ×⎣ ⎦
+ − + ×
− − + ×
∫ ∫
∫ ∫
∫
∫
(19)
The second term of Eq. (16) is the potential energy corre-
sponding to the tension of a cable. Similarly to the kinetic energy, rearranging it can be expressed as Eq. (20).
( )( )( )( ) ( )
2
1
2 2
1 1
0
0 0
t L
s st
L t t L
s st ts
T s R R dsdt
T s R Rdtds T s R R dt
δ
δ δ
⋅
⎡ ⎤= − + ⋅⎣ ⎦
∫ ∫
∫ ∫ ∫ (20)
The third and fourth terms of Eq. (16) are the virtual work
by fluid resistance forces and the virtual momentum transport across the open boundary, respectively. So, rearranging them can be expressed as Eqs. (21) and (22).
( )2
1 0,
t L
atF s t Rdsdtδ⋅∫ ∫ . (21)
( ) ( )( )2
1 0
lst
tAV DR R R dtρ ω δ⎛ ⎞+ × ⋅⎜ ⎟
⎝ ⎠∫ . (22)
Substituting Eqs. (19)-(22) into the extended Hamilton’s
principle, Eq. (16) can be rearranged as Eq. (23).
( ) ( )( )( )( ) ( )
( )( )( )
2
1
2
1
2
1
2
1
2
0
0
0
0
2
,
0l
L t
t
L t
s at s
L t
t s t
t s
st
A D R DR R Rdtds
T s R F s t Rdtds
A R VR R R ds
T s R R dt
ρ ω ω ω δ
δ
ρ ω δ
δ
⎡ ⎤− + × + × ×⎣ ⎦
⎡ ⎤− − −⎣ ⎦
+ − + ×
− ⋅ =
∫ ∫
∫ ∫
∫
∫
(23)
Because the virtual displacement 0Rδ = at arbitrary time
1290 J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
1 2, t t and Eq. (23) must be satisfied for all virtual displace-ments, Rδ within every region 0 ls s< < , the governing equations of motion and the boundary conditions at two points can be derived as Eqs. (24) and (25), respectively.
( ) ( )( )( )( ) ( )
22 2
,
tt ts ss t s
s as
A R VR V R R VR R
T s R F s t
ρ ω ω ω− + + × − + × ×
= + (24)
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
1
2
0, 0 or 0, at eyelet
, 0 or , at lift-off points
s l l
T s R t R t R t
T s R s t R s t R t
= =
= = (25)
where ( ) ˆ ˆ, r zR s t re ze= + is the position vector of point P which is defined from the cylindrical coordinates at guide eyelet.
The physical meanings of terms in Eq. (24) are as follows. The first term is the local inertial force of cables due to the translational unwinding velocity and the second and third terms are the Coriolis and centrifugal forces caused by transla-tional unwinding velocity and deformation of cables, respec-tively.
The fourth term is the Coriolis force due to the coupling of translational and rotational unwinding velocity, and the final term is the centrifugal force due to rotational unwinding ve-locity.
By systematically deriving the equations of motion by using the extended Hamilton’s principle for open systems, the ex-tension into unwinding of cables which have a bending stiff-ness or very thick cables which have a rotary inertia and shear deformation effect is possible just through simply modifying the kinetic and potential energy terms, Eqs. (10) and (11). 2.4 Fluid-resistance forces
Because unwinding cables from cylindrical packages undergo nonlinear large displacements in spite of small deformations, the effect of fluid-resistance forces is dominant to behaviors of unwinding cables.
Fig. 4 shows the velocity vector of a cable element during unwinding. At arbitrary time t , the velocity vector of a cable element due to translational and rotational unwinding velocity is written as Eq. (26).
v DR Rω= + × . (26) The normal component of the velocity can be expressed as
Eq. (27) containing a triple cross product.
( ) ( )( )( )
n s s s s
s s
v v v R R R v R
R DR R Rω
= − ⋅ = × ×
= × + × × (27)
Using normal velocity of a cable element written in Eq. (27),
the fluid-resistance force can be expressed as the following [8]:
1F2a D w n n n n nC d v v D v vρ= − = − , (28)
where wρ is the density of surrounding fluids, d is the effective diameter of a cable, and DC is the drag coefficient of a cable element. In Eq. (28), the minus sign means that fluid-resistance forces act in the opposite direction to the motion of the cables. So, this leads to 20.5 4.27 4 kg/mn D wD C d Eρ= ≈ − in the case that a cable’s diameter is 0.7mm and the cable is surrounded by air. 2.5 Inextensible conditions of cables
As shown in the potential energy in Eq. (11), the restoration force of tensile forces is only included in governing equations of motion of unwinding cables. Namely, it is assumed that the cable used in this study is perflectly flexible, uniform, inextensible to the longitudinal direction, and irresistible to bending forces. So, an additional equation which is inextensible to the longitudinal direction of the cable is required in governing equations of motion. The inextensible conditions of cables can be wriiten as Eq. (29).
2 2 2 2 1s s s s sR R r r zθ⋅ = + + = . (29)
Differentiating Eq. (29) by the time and space variable, Eqs. (30) and (31) are obtained, respectively.
( ) ( ) ( )2
ˆ ˆ ˆ
ˆ ˆ ˆ2s s r s s z
ss ss s r ss s s ss z
R r e r e z e
R r r e r r e z eθ
θ
θ
θ θ θ
= + +
= − + + + (30)
( ) ( ) ( )( ) ( ) ( )
2
ˆ ˆ ˆ
ˆ ˆ ˆ2
ˆ ˆ ˆ
t t r t t z
tt tt t r t t tt tt z
ts ts s t r s t t s ts ts z
R re r e z e
R r r e r r e z e
R r r e r r r e z e
θ
θ
θ
θ
θ θ θ
θ θ θ θ θ
= + +
= − + + +
= − + + + +
(31)
ω
ˆreeθ
ˆze
O′rC
P
V
O
LL
v
sv R⋅
( )s sv R R⋅
v
P
nv
sR
( )n s sv v v R R= − ⋅
Fig. 4. The normal velocity of a cable element when the cable is un-winding.
J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296 1291
3. Numerical analysis of unwinding cables
3.1 Governing equations on cylindrical coordinates
The non-linear governing equations of unwinding motion which are derived in section 2.3 were written again in Eq. (32).
( ) ( )( )( )( ) ( )
22 2
,
tt ts ss t s
s as
A R VR V R R VR R
T s R F s t
ρ ω ω ω− + + × − + × ×
= + (32)
If this vector equation is changed into scalar forms with
respect to the cylindrical coordinates attached at the guide eyelet, three equations are derived as Eqs. (33)-(35).
2
2 2 2 2
2
2 2 2
2
T T T
tt ts t t s t
ss s s
ss s s s r
mr mVr m r mr mVr
mV r mrV mr V mr
r r r F
ω θ θ θ θ
θ ω θ ω
θ
− − − +
+ − + −
= − + +
(33)
2 2
2 2 2 2
2 2 2 T 2T T
tt t t s t ts t s
t ss s s s
ss s s s s
mr mr mVr mrV mVr
m r mrV mV r m Vrr r r Fθ
θ θ θ θ θ
ω θ θ ωθ θ θ
+ − − −
+ + + −= + + +
(34)
22 T Ttt ts ss ss s s zmz mVz mV z z z F− + = + + , (35) where 0 ls s≤ ≤ and ls is the arc-length measured along to the length of the cable from the guide eyelet to the lift-off point. The differential operators are ( ) ( )2 2/
ttt≡ ∂ ∂ ,
( ) ( )2 /ts
t s≡ ∂ ∂ ∂ and ( ) ( )2 2/ss
s≡ ∂ ∂ . , , r zF F Fθ which are included in Eqs. (33)-(35), mean fluid-resistance forces along three directions of the cylindrical coordinates, ˆ ˆ ˆ, , r ze e eθ . Fluid resistance forces are calculated as Eq. (36).
( ) ( ) ( )
12
ˆ ˆ ˆ
a D w n n n n n
r r z z
F C d v v D v v
F e F e F eθ θ
ρ= − = −
= + + (36)
where DC is the drag coefficient that is a function of the Reynolds number, wρ is the density of surrounding fluid, d is the effective diameter of the cable, and nD is the coeffi-cient of fluid-resistance.
nv means the normal velocity with respect to the motion of the cable and is calculated as Eq. (37) in cylindrical coordi-nates that is located at guide eyelet. Inserting the normal ve-locity nv into Eq. (36), the fluid-resistance force vectors can be calculated and the governing equations of motion in cylin-drical coordinates can be completed.
( )( )( )( )
2 2 2 2 2
2 2 2 2
2 2 2 2 2
R R
ˆ
ˆ
ˆ
n s s
t s t s s s s t s s t s r
t s t s t s s s t s s s
t s t s s t s s t s s s z
v v
r r r r r r z r z r z e
r z r r rr r r r rz z r z e
z r r z r r z r z r z e
θ
θ θ θ ω θ
θ θ θ ω θ ω
θ θ θ ω θ
= × ×
= − − − +
+ + − + − +
+ − − + −
(37)
In this paper, it is assumed that the tension only depends on
the arc length coordinates of the cable and only acts along the cable axis. So, by applying the dot product of R s at both sides of the governing equations, Eq. (32), the tension equation is calculated as Eq. (38).
2s sT mr rω= − (38)
Integrating Eq. (38) with respect to the whole balloon length,
the tension equation at the position level can also be calculated as Eq. (39).
22
0 2mT T rω
= − (39)
where 0T means the tension at the guide eyelet. So, if one knows the tension at the guide eyelet 0T , the tension distribu-tion of the whole balloon can be calculated and the balloon behavior can be solved by using Eqs. (33)-(35).
By substituting Eqs. (38) and (39) into Eqs. (33)-(35), the governing equations of unwinding cables represented in cy-lindrical coordinates can be calculated as Eqs. (40)-(42).
( )
2
2 22 2 2 2
0
2
2 2 2 +2
2
0
tt ts t t s t s
s ss s
r
mr mVr m r mr mVr mr V
mrmr r mV T r r
mr F
ω θ θ θ θ ω θ
ωω θ
ω
− − − +
⎛ ⎞+ + + − −⎜ ⎟
⎝ ⎠− − =
(40)
( )2 2
20
2 2
2 2 2 2 2
2 22
0
tt t t s t ts t s t
s ss s s
s s
mr mr mVr mrV mVr m r
mrm Vr mV T r r
mr r Fθ
θ θ θ θ θ ω
ωω θ θ
ω θ
+ − − − +
⎛ ⎞− + + − +⎜ ⎟
⎝ ⎠+ − =
(41)
2 22
0
2
22
0
tt ts ss
s s z
mrmz mVz mV T z
mr r z F
ω
ω
⎛ ⎞− + + −⎜ ⎟
⎝ ⎠+ − =
(42)
Because the fluid-resistance force is already known as Eq.
(36), the behavior of cables from the cylindrical coordinates can be solved if the tension at the guide eyelet is known. In this paper, to solve these equations numerically, the modified finite difference methods are suggested.
3.2 Two points boundary conditions
Balloon behavior of unwinding cables is determined within from a lift-off point to the guide eyelet. So, boundary condi-tions must be determined at these two points.
When two boundary conditions written in Eq. (25) are transformed into a function of time variable, those can be re-solved into boundary conditions at a guide eyelet and lift-off point, as shown in Eqs. (43) and (44).
( ) ( )( ) ( )
0, 0, 0, 0
0, 0, 0,t t
r s t z s t
r s t z s t V
= = = =
= = = = − (43)
1292 J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
( ) ( ) ( )( ) ( ) ( )( )
, , , , ,
, 0, , , , 0
, 0
l r l l B
t l t l t l
s l
r s s t C s s t t z s s t H
r s s t s s t z s s t
r s s t
θ ω
θ ω
= = = = = =
= = = = = =
= =
(44)
where BH and rC are the balloon height and the radius of cylindrical package, respectively. The rotational unwinding velocity / rV Cω = is calculated as in the case of a zero winding angle. At a guide eyelet of Eq. (43), ( )0,s tθ = and ( )0,t s tθ = are the boundary conditions that must be deter-
mined according to compatibility conditions during solving governing equations. The term ( ), 0s lr s s t= = means the geometrical constraint that says no radial slope at a lift-off point due to the cylindrical packages, as shown in Fig. 5.
In this study, since the control volume is set as the balloon region which consists from a guide eyelet to lift-off point and no moving of the lift-off point is assumed during unwinding, the constraint, ( ), 0t lz s s t= = , is added.
3.3 Modified finite difference methods
The governing equations of unwinding cables represented in cylindrical coordinates, Eqs. (40)-(42), are non-linear cou-pled partial differential equations with respect to the time vari-able t and spatial variable s . So, the discretization of the governing equation is very complicated. In this study, to avoid tedious treating non-linear and coupled terms at each time step, the modified finite difference method is added into the central difference method at the discretization of spatial variables. And the Newmark integration method [6, 7] is used in the time marching.
Dividing spatial variables using the central difference method which has second order accuracy of 2O s⎡ ⎤∆⎣ ⎦ , the differences could be expressed as Eq. (46).
, 1,2,...is i s i n= ∆ = , (45)
( )2
1 1 1 12 2
1 1
2 , 2
2
i i i i iss st
i is
r r r r r rr r rs s s sr r rrs s
+ − + −
+ −
∂ − + ∂ −= = = =∂ ∆ ∂ ∆∂ −
= =∂ ∆
(46)
where ( ) / t
⋅
= ∂ ∂ and other spatial variables, θ and z , can also be discretized using similar way. Substituting discretized spatial variables into the governing equations, the following equations of motion can be obtained;
( ) ( ) [ ]
( ){ } , 0
t t t t t t t t t t
t t t t
M q q C q q K q
D q q
+∆ +∆ +∆ +∆ +∆
+∆ +∆
⎡ ⎤ ⎡ ⎤+ +⎣ ⎦ ⎣ ⎦
+ = (47)
where t t+ ∆ means the precedent time stage which the equations of motion is derived and [ ], , ,T
i i i iq r zθ= ( )2 ~i n= is the vector of state variables and the total size of
the given system is ( )3 1 1n − × . Mass matrix ( )M q⎡ ⎤⎣ ⎦ , quasi-damping ( )C q⎡ ⎤⎣ ⎦ , and quasi-stiffness matrix [ ]K , and nonlinear vector ( ){ },D q q are defined as Eqs. (48)-(50), respectively.
[ ]6
6
6 3 9
0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0
ei i
CM C r
C×
⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦
(48)
[ ] [ ]1 2 3 3 9, ,ei e e eC C C C
×=
where
[ ] [ ]
[ ]( )( )
( )( ) ( )
1 1
1 1 3 1
1 13 3 3 3
1 1 1 2
2 2 1 1 1 1 1 1
3 3
0 0 0 00 0 , 0 00 0 0 0
0 0
0
0 0 0
e i e i
i i i
e i i i i
C CC C r C C r
C C
C C r
C C C C r r
θ θ
θ θ
× ×
+ −
+ − + −
×
−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦
⎡ ⎤− −⎢ ⎥
= − − − −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
(49) [ ] [ ]1 2 3 3 9
, ,ei e e eK K K K×
=
where
[ ] [ ]( )
[ ]
3 3 5
1 4 2
3 33 3 3 3
3
3 4
3 3 3
0 0 2 0 00 0 , 0 0 0
0 0 0 0 2
0 00 0
0 0
e e
e
C C CK C K
C C
CK C
C
× ×
×
− ⎡ − ⎤⎡ ⎤⎢ ⎥⎢ ⎥= = ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥−⎣ ⎦ ⎣ ⎦
−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥−⎣ ⎦
(50) { } { }
( ) ( )
( ) ( )
( )
( )( ) ( )
3 1
2 231 1 4 1 1 6
2 237 71 1 1 1
271 1
31 1 1 1 3 1 1
376 1
, ,
4
8 4
22
22
2 22
Tei i i i
i i i i i i i i i
i i i i i i
i i i i r
i i i i i i i i i
i i i i
D G Q W
CG r C r C r
C Cr r r r
C r r r r F
CQ r r C r
CC r r
θ θ θ θ θ
θ θ
θ θ θ θ θ
θ θ
×
+ − + −
+ − + −
+ −
+ − + − + −
+
=
≡ − + − −
− − + −
+ − + −
≡ − − − − − +
+ + −( )
( )( )
1
271 1 1 1
2
i i
i i i i iC r r r Fθ
θ θ
θ θ
−
+ − + −
+
+ − − −
Guide eyelet
⑦ ⑧ ⑨
Lift-off point
Top view
⑨ Lateral view
0ls s
rs =
∂=
∂
Fig. 5. Graphical descriptions of geometrical constraints, ( ) 0.s lr s =
J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296 1293
( ) ( )( )27 71 1 1 1 1 12
2 4i i i i i i i i i i zC CW r z z z r r r z z F+ − + − + −≡ − + + − − −
(51)
where, constants of Eqs. (48)-(51) are same as Eq. (52).
20
1 2 3 2
22
4 5 6 7 2
8 2
, 2 ,
, , ,
14
mV T mVC C m Cs s
m V mC C m C m Cs s
Cs
ω
ω ωω
−= = =
∆ ∆
= = = =∆ ∆
=∆
(52)
3.4 Newmark time integration method
To solve time responses of unwinding cables, Eq. (47), the Newmark time integration method [6, 7] is applied in this study. The Newmark time integration involves implicit time integration and gets solutions using precedent equilibrium equations. Because non-linear equations must be solved with an implicit time integration method, Newton-Raphson itera-tions are applied in this study with the following relations:
( )
2
1
12
t t t t t t
t t t t t t t
q q q q t
q q q t q q t
δ δ
α α
+∆ +∆
+∆ +∆
⎡ ⎤= + − + ∆⎣ ⎦⎡ ⎤⎛ ⎞= + ∆ + − + ∆⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦
(53)
where α and δ are parameters that can be determined to obtain integration accuracy and stability. When 0.25α = and
0.5δ = , Eq. (53) corresponds to the constant acceleration method and guarantees unconditional stability. Rearranging Eq. (53) for the term of t tq +∆ , then the following equations could be obtained:
( )2 2
0.51 1 1
2 12
t t t t t t t
t t t t t t t
q q q q qt t t
tq q q q qt t
αα α α α
δ δ δ δα α α α
+∆ +∆
+∆ +∆
−= − − −
∆ ∆ ∆∆⎛ ⎞ ⎛ ⎞= − + − + −⎜ ⎟ ⎜ ⎟ ∆ ∆⎝ ⎠ ⎝ ⎠
(54)
( ) ( )
( ) ( ) [ ]0 0 2 3
5 4 1 1
t t t t t t t
t t t t t t t t t
M q a q a q a q a q
C q a q a q a q a q K q
+∆ +∆
+∆ +∆ +∆
⎡ ⎤ − − −⎣ ⎦⎡ ⎤+ − − + − +⎣ ⎦
( ) , 0t t t tD q q+∆ +∆⎡ ⎤+ =⎣ ⎦ (55)
( )
0 1 22
3 4 5
6 7
1 1, ,
1 1, 1, 2 .2 2
1 ,
a a at t t
ta a a
a t a t
δα α α
δ δα α α
δ δ
= = =∆ ∆ ∆
∆ ⎛ ⎞= − = − = −⎜ ⎟⎝ ⎠
= ∆ − = ⋅ ∆
(56)
Eq. (55) is the time varying system in which system matri-
ces and vectors are changed at every time step and non-linear equations in which all t tq +∆ are unknown except both
boundary conditions. So, to get the solution of Eq. (55), New-ton-Raphson iteration is applied. Using Newton-Raphson iteration, t tq +∆ could be obtained to satisfy equilibrium equa-tions at every time step. t tq +∆ and t tq +∆ can be also solved using Eq. (54).
4. Unwinding behaviors of cables
Using the modified finite difference method and Newmark time integration scheme explained in section 3, unwinding behaviors of cables were analyzed. The dimensions and un-winding parameters of the used cylindrical package are listed in Table 1 and in Fig. 6. A cable with a density of 5.67E-6 kg/m is drawn through the guide eyelet with constant velocity 600 m/min from the cylindrical package. The radius and length of the package are 0.123 m and 0.400 m, respectively. And the balloon height BH , which means the distance from a guide eyelet to a lift-off point, is 0.450 m.
Two unwinding balloon shapes according to two different tensions, 0.45 cN and 0.22 cN at a guide eyelet, are shown. Because tension is one of the most significant factors of the unwinding motion, above two tensions at a guide eyelet make the different unwinding behaviors in spite of the small differ-ence. 4.1 Result with a higher tension
Unwinding results of cables from the cylindrical package with the tension of 0.45 cN are shown in Figs. 7-10. In Fig. 7, top and lateral views of unwinding balloon shapes are drawn.
At the initial position, the cable has tension of 0.45 cN and is positioned in a straight line. When pulling with an unwind-ing velocity of 600 m/min through the guide eyelet, Fig. 7 clearly demonstrates the unwinding sequence. Since the ten-sion 0.45 cN is comparatively higher, the cable is unwinding with only single balloon shape.
And as shown in Fig. 8, the transition states from the initial straight cable to the single balloon can be checked step by step. Because the lift-off point is fixed, one can imagine that the
Table 1. Unwinding parameters of a cylindrical package.
Mass ( m )
Velocity (V )
Radius ( rC )
Balloon height ( BH )
Length ( l )
[kg/m] [m/min] [m] [m] [m]
5.67E-6 600.0 0.123 0.450 0.400
0.450 mBH =
0.123 mrC =
0.40 ml =
600 m/minV =
Fig. 6. Dimensions and unwinding parameters of a used cylindrical package.
1294 J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
unwinding behaviors are stabilized. However, the cable is still oscillating according to the spin motion of the lift-off point shown in Fig. 9. In the first plot of Fig. 9, the balloon ampli-tude means the distance which is apart from the axis of the cylindrical package. The second plot of Fig. 9, oscillating
angle means the rotating angle minus the angle of the lift-off point. Vertical displacement means the cable’s displacement along the package axis. Obviously, these three plots mean displacements in the cylindrical coordinates. The vertical dot-ted line in Fig. 9 indicates the time for one complete cycle of the spin movement at the lift-off point. In Fig. 9, although the first two curves are stabilized within 1 cycle, the vertical dis-placement of the cable is still oscillating. So, it is difficult to say that the unwinding motion has reached a steady state. In Fig. 10, one can see the unwinding motion of cables according to the time marching.
4.2 Result with a lower tension
Unwinding results of cable with a lower tension of 0.22 cN are shown in Figs. 11-14. In the case of 0.22 cN, which is comparatively smaller, the cable is unwinding with a double balloon shape. As shown in Fig. 12, the motion starts from the initial straight cable, and passing to a single balloon, finally proceeds to a double balloon, which can be checked in Fig. 12.
As shown in Fig. 13, although the first two curves are stabi-lized within 1 cycle, the vertical displacement of the cable is still oscillating as the same as the higher tension. In Fig. 14, one can also see the unwinding motion of cables in the case of 0.22 cN according to the time marching. One can see that the cable’s motion is more flexible and has more balloons due to lower tension.
5. Conclusions
To prevent unwinding problems of a cable, such as tangling or cutting, unwinding behaviors of cables from cylindrical
0.05
0.1
0.15
30
210
60
240
90
270
120
300
150
330
180 00ms
6ms
42ms
78ms60ms
24ms
-0.2 -0.1 0 0.1 0.2
-0.2
0
0.20
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x displacementy displacement
z di
spla
cem
ent
0.0ms18ms
48ms
Fig. 7. Unwinding balloon shapes in the case of 0.45 cN at a guideeyelet (single balloon).
0.0 0.2 0.4
0.00
0.05
0.10
0.15
0.20
Ballo
on R
adiu
s [m
]
Balloon height [m]
t = 0 [ms] t = 6 [ms] t =12 [ms] t =18 [ms]
0.0 0.2 0.4
Balloon height [m]
t= 24 [ms] t= 30 [ms] t= 36 [ms] t= 42 [ms]
(a) 0-18 ms (b) 24-42 ms
0.0 0.2 0.4
0.00
0.05
0.10
0.15
0.20
Bal
loon
Rad
ius
[m]
Balloon height [m]
t= 48 [ms] t= 54 [ms] t= 60 [ms] t= 66 [ms]
0.0 0.2 0.4
Balloon height [m]
t= 72 [ms] t= 78 [ms] t= 84 [ms] t= 130 [ms]
(c) 48-66 ms (d) 72-130 ms Fig. 8. Transition of unwinding balloon shapes from a fixed coordinateat a lift-off point until 130 ms (single balloon).
0.00 0.02 0.04 0.06 0.08 0.10 0.120.215
0.220
0.225
0.230
0.235-1.0
-0.8
-0.6
-0.4
-0.2
0.00.06
0.08
0.10
0.12
0.14
0.16
0.18
Verti
cal
Dis
plac
emen
t [m
]
T ime [s]
Osc
illatin
gAn
gle
[rad]
Bal
loon
Ampl
itude
[m]
Fig. 9. Displacement of a center node in the case of 0.45 cN, a verticaldashed line means 1 cycle of unwinding.
(a) t = 0 ms (b) t = 20 ms
(c) t = 33 ms (d) t = 49 ms
(e) t = 59 ms (f) t = 77 ms
(g) t = 90 ms (h) t = 130 ms Fig. 10. Unwinding balloon shapes according to time marching in the case of 0.45 cN (single balloon).
J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296 1295
spool packages are derived and analyzed. First, the governing equations of motion of cables unwinding from a cylindrical spool package are systematically derived using the extended Hamilton’s principles for an open system, which mass can be transported at each boundary. To numerically solve the derived
governing equations, the modified finite difference method was applied. Time responses of unwinding cables are calculated using the Newmark time integration method. For the two cases of tension at guide eyelet, the unwinding behaviors of cables are studied and compared. It was verified that balloon shapes were changed depending on the tension values. From the simu-lation results, it is shown that one can predict the unwinding motion of cables from cylindrical packages analytically.
In this paper, since the lift-off point is assumed fixed during unwinding, the control volume is fixed as the balloon region, which is not realistic. Thus, our next research is to analyze the unwinding motion in the case when the lift-off point is moving along the packages axis. In that case, the relationship between the tension of the cable and the friction force between cables and packages should also be considered.
Acknowledgment
This work was supported by Agency for Defense Develop-ment (ADD) and Mid-Career Researcher Program through NRF grant funded by the MEST (NO. 2010-0027555).
References
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[2] W. B. Fraser, T. K. Ghosh and S. K. Batra, On Unwinding Yarn from a Cylindrical Package, Proceedings: Mathemati-cal and Physical Sciences (436) 1898 (1992) 479-498.
[3] Wire Guided Missle, MILAN Project, http://www.eliteuk-forces.info/weapons/milan/.
[4] Wire Guided Torpedo, DM2A4, http://www.atlas-elek-tronik.com/.
-0.2
0
0.2
-0.2-0.15-0.1-0.0500.050.10.150
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
x displacy displacement
z di
spla
cem
ent
0.05
0.1
0.15
30
210
60
240
90
270
120
300
150
330
180 00ms
78ms
42ms0.0ms
78ms
18ms
Fig. 11. Unwinding balloon shapes in the case of 0.22 cN at a guideeyelet (double balloon).
0.0 0.2 0.4
0.00
0.05
0.10
0.15
0.20
Bal
loon
Rad
ius
[m]
Balloon height [m]
t = 0 [ms] t = 6 [ms] t =12 [ms] t =18 [ms]
0.0 0.2 0.4
Balloon height [m]
t= 24 [ms] t= 30 [ms] t= 36 [ms] t= 42 [ms]
(a) 0-18 ms (b) 24-42 ms
0.0 0.2 0.4
0.00
0.05
0.10
0.15
0.20
Ballo
on R
adiu
s [m
]
Balloon height [m]
t= 48 [ms] t= 54 [ms] t= 60 [ms] t= 66 [ms]
0.0 0.2 0.4
Balloon height [m]
t= 72 [ms] t= 78 [ms] t= 84 [ms] t= 130 [ms]
(c) 48-66 ms (d) 72-130 ms Fig. 12. Transition of unwinding balloon shapes from a fixed coordi-nate at a lift-off point until 130 ms, 0.25 cN (double balloon).
0.00 0.02 0.04 0.06 0.08 0.10 0.12
0.22
0.24
0.26
0.28-2.0
-1.5
-1.0
-0.5
0.0
0.06
0.08
0.10
0.12
0.14
Ver
tical
D
ispl
acem
ent [
m]
T ime [s]
Osc
illat
ing
Angl
e [ra
d]B
allo
onAm
plitu
de [m
]
Fig. 13. Displacement of a center node in the case of 0.25 cN, a verti-cal dashed line means 1 cycle of unwinding.
(a) t = 0 ms (b) t = 11 ms
(c) t = 20 ms (d) t = 40 ms
(e) t = 60 ms (f) t = 70 ms
(g) t = 98.2 ms (h) t = 120 ms Fig. 14. Unwinding balloon shapes according to time marching in the case of 0.22 cN (double balloon).
1296 J.-W. Lee et al. / Journal of Mechanical Science and Technology 25 (5) (2011) 1287~1296
[5] D. B. McIver, Hamilton’s Principle for Systems of Changing Mass, Journal of Engineering Mathematics, (7) 3 (1973) 249-261.
[6] N. M. Newmark, A Method of Computation for Structural Dynamics, ASCE Journal of Engineering Mechanics Divi-sion (85) (1959) 67-94.
[7] K. J. Bathe, Finite Element Procedures, Prentice Hall Inter-national, Inc., (1996).
[8] The AUTOFLUG Product Range Measuring and Sensor Technology, http://www.autoflug.com.
[9] P. M. Gerhart, R. J. Gross and J. I. Hochstein, Fundamentals of Fluid Mechanics, 2nd edition, Addison-Wesley Publish-ing Company (1992).
[10] J. Kiusalaas, Numerical Methods in Engineering with Mat-lab, Cambridge University Press (2005).
Jae-Wook Lee received his B.S. degree from Pusan National University (1999), M.S. from Pusan National University (2001), and Ph.D. from Pusan National University (2010). He is currently a full time research engineer at Research Insti-tute of Mechanical Technology in Pusan National University. His major area is
flexible multi-body dynamics based on FEM and non-linear dynamics using numerical methods.
Deuk-Man An received his B.S. degree from Seoul National University (1977), M.S. from KAIST (1979) and Ph.D. from Rensselaer Polytechnic Institute (1986). He is currently a Professor at the School of Mechanical Engineering at Pusan National University. His major area is fracture mechanics and contin-
uum mechanics.
Wan-Suk Woo received his B.S. degree from Seoul National University (1976), M.S. from KAIST (1978) and Ph.D. from University of Iowa (1985). He is currently a Professor at the School of Mechanical Engineering at Pusan Na-tional University, and serving as a presi-dent for the KSME in 2011. His major
area is vehicle dynamics and flexible multibody dynamics.
