finite size effects on twisted rods stability - ida.upmc.fr · • semi -finite correction. •...
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Finite Finite size effects on twisted rods stabilitysize effects on twisted rods stabilitySébastien NeukirchSébastien Neukirch (joint work with G. van (joint work with G. van derder HeijdenHeijden & J.M.T. Thompson) & J.M.T. Thompson)
Elasticity of twisted rods
supercoiling of DNA plasmids
carbon nanotubesApplications
sub marine cables Climbing plants
2 types of experiments
Hypothesis
• no shear
• no extensibility
• no intrinsic curvature
• no gravitation
• rotation without sliding (D constant)
• sliding without rotation (constant R)
Equilibrium equations
u
GJ
EI
EI
M
duddS
d
dRdS
d
dFMdS
d
FdS
d
ii
rr
rrr
rr
rrr
r
=
×=
=
×=
=
00
00
00
0
3
3
)(
)(
)(
)(
)(
)(
)(
3
2
1
Su
Sd
Sd
Sd
SR
SM
SF
r
r
r
r
r
r
r
• Boundary conditions :
• 1 independent variable S : ODEs
• Static-Dynamic Kirhhoff analogy : spinning top <=> spatial elasticaelastica
7 unknowns
)()()( 333 BdAdBdkBArrrr
==
integrable
21 EIEI =
7 equations
What do we want to get ?
• All the static configurations of the rodfor the clamped boundary conditions.
• Stability of these configurations underthe 2 typical experiments.
Systems of ODEs with :
initials conditions
boundary conditions
In the parameters space ( L , EI , F, M , ...) there will be a ‘n-D’ solution manifold.
Define an index I [K. Hoffman]:I = 0 : stable,I = 1 : unstable,I > 1 : more unstable.
Number of negative eigenvalues of the second order differentialoperator of the constrained variational problem.
3 different models
∞=L
∞≤L
finiteL
homoclinic trajectory
homoclinic trajectory without fixed point
other trajectories in phase space
A
C
B
• A & B are much easier than C because less parameters
• we will compare, as we go A → B → C, :
1 - how stability changes,2 - how new solutions appear.
• Reduction to an equivalent oscillator (2D)
• Spatial localisation of the deformation.
• Applied force and moment // rig axis.
• Buckling :
• D end-shortening
• Parameters space :
• Solution manifold :
Rod of infinite length (van der Heijden - Champneys - Thompson)
TM 42 =
−=
T
M
TD
41
16 2
∞=R (as soon as M > 0)
( )( )max2 cos12 θ+= TM
Rigid Loading (R,D) :
max,, θTM
Rod of infinite length : sliding without rotationsliding without rotation
E
F
G
H
2max
πθ =1 2 3 4 5
M1
2
3
4
5
6
7
8T
unstable stable
this is a projection !
Very long rod : constconst. R. R const const. M. M≠
• Length L :
• Same formula for D :
−=
t
m
td
41
16 2
• Same solution manifold : ( )( )max2 cos12 θ+= tm
]2
1;
2
1[,,,,
2
+−∈===Γ==L
Ss
L
Dd
EI
GJ
EI
LTt
EI
LMm
• because we consider only part of the homoclinic :
• R depends on Γ whereas d and m2 do not.
+
Γ=
t
mArcCos
mR
24
∞<R
[Heijden, Thompson]
Very long rod : post-buckling surfacepost-buckling surface
1 2 3 4 5m
1
2
3
4
5
6
7
8t
• Semi-finite correction.
• tangency between const. R and const. D curves
• Stability limit :
- corresponds to
- more stable than in the infinite case.
( )
+Γ+Γ−=
t
ttm
4
214
22
2
• Special curve :
2max
πθ >
π2=R
• No instability for : π2<R
unstable stable
0.5 1 1.5 2 2.5 3 3.5 4 d
2
4
6
8
t
Very long rod : stability for constant stability for constant RR experiments experiments
• Stability changes at folds in D.
• We tune D : conjugate parameter is T.
• Index [Maddocks] ; potential V [Thompson] :
Level curves of :
−+−
Γ=
1614
161
2 22
0
tdArcCos
tdtR
• Special path : π2=R
• Curves with have no foldπ2<R
ddt
Γ+=
482
Lift-off Loss of stability :
0R
2 4 6 8 10 12R
1
2
3
4
5
6
7
8m
Very long rod : stability for constant stability for constant DD experiments experiments
• Stability changes at folds in R
• We tune R, conjugate parameter is - m
−±+
Γ=±
02
0
164224
dm
dmArcCos
mR
Level curves :
π22
+
Γ
−Γ
=m
ArcTanm
R
• At low D : jump to contact.
• At large D : quasistatic to planar elastica.
Γ=
2MILd
Loss of stability :
0d
Finite length rod : homoclinic
• Buckling under compression.
• Same equations as before but more (free) parameters.
• System is still integrable.
• No homoclinic trajectory in phase space (generically).
• Equivalent Wrench (M, F) no longer // rig.
• Shear force and bending moment at the clamps.
θ
θ
Finite length rod : buckling modes (clamped)buckling modes (clamped)
−=−
−
− tmSinttm
mCostmCos 4
2
14
24
2
1 222
−4 −2 2 4mz
−4
−3
−2
−1
1
2
3
4t
∞
1st
2nd
3rd
2
2
=
πL
B
Tt
π2
L
B
Mm z
z =
Finite length rod : planar modes (clamped)planar modes (clamped)
−4 −2 2 4 6t
1
2
3
4
mx0
∞
1st inflex
3rd inflex
1st non-inflex
2nd non-inflex
3rd non-inflex
2
2
=
πL
B
Tt
π20
0
L
B
Mm x
x =
Finite length rod : where is the post-buckling surface ?where is the post-buckling surface ?
01
23
4mz
−4
−2
02
4t
0
1
2
3
4
mx0
01
23
4mz
−4
−2
02
4t
buckling 1st mode
non-inflex 1st mode
inflex 1st mode
• we consider :
- non trivial initial conditions :
- free parameters :
• Global Representation Space [Domokos]
<=> unique configuration.
Boundary conditions :
• 2D solution manifold in 4D space.
• All the non-closed configurations
are s-symmetric (because ).
Finite length rod : post-buckling surfacepost-buckling surface
z
y
x
yxxyz
xzzxzy
yzzx
dz
dy
dx
dmdxtdytd
dmdmdytd
dmdxtd
3
3
3
30333
33033
333
=′
=′=′
+−−=′
+−=′
−=′
0θ
0,, xz mtm
( )00 ,,, θxz mtm
=
=
xz
y
dzdx
d
33
3 0
• Boundary value problem (BVP).
• Centre line of rod : 6D system :
00 =ym
Finite length rod : sliding without rotationsliding without rotation
LIFT OFF
BIRD CAGE
0.2 0.4 0.6 0.8 1 1.2 1.4D
−4
−3
−2
−1
1
2
3
t
R=7 rad .
0.2 0.4 0.6 0.8 1 1.2 1.4D
−3
−2
−1
1
t
R=6 rad .
0.2 0.4 0.6 0.8 1 1.2 1.4D
−2.5
−2
−1.5
−1
−0.5
0.5t
R=1 rad .
0.2 0.4 0.6 0.8 1 1.2 1.4D
−2
−1
1
2
3
t
R=10 rad .
Finite length rod : 3 typical shapes3 typical shapes
0 < D < 1 D = 1 1 < D < 0
closed revertedopened
Finite length rod : rotation without slidingrotation without sliding
Same fold as before (jump to contact).
New fold leading to hysteresis
D0=0.5 D0=0.8
D0=0.9
2π
-2π
R
M
WX
YZ
O
D0=0.6
D0=0.7 D0=0.99−2 π 2 π
R
−1
1
m3
Finite length rod : connection between 1st and 2nd buckled modesconnection between 1st and 2nd buckled modes
−4−2
02
4
mz0
12
34
t
0
1
2
3
4
mx0
−4−2
02
4
mz
0
1
2
3
4
mx0
D = 1.1planar 1st mode inflex planar 2nd mode non-inflex
−2 π −π π 2 πR
−0.5
0.5
1
m3
Finite length rod : how to change how to change Γ
Γ2
Γ1
∆ ∆
Comparison of the 3 models for the jump to contactjump to contact
DêD∗
2 π
4 π
6 π
8 π
R
∞=L
∞≤L
finiteL
Finite length rod : overall stabilityoverall stability
1 2D
2 π
4 π
6 π
8 π
R
contact
hysteresis hysteresis
Planar elastica : connections between connections between inflexinflex. paths. paths
05
10t
0
πθ
0
2
4
6
mx0
05
10t
0
πθ