1. No category

Dynamique d’une chaˆıne d’aimants
Application `
a la r´
ecup´
eration d’´
energie
M. Ducceschi(a) , C. Rouby(a) , L. Bodelot(b)
(a) UME - ENSTA ParisTech
(b) LMS - Polytechnique
RFM, Fontainebleau
12 - 05 - 2015
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
1 / 37
Energy Harvesting
Aim: to harvest the energy of a fluid flow
Dynamical instability leads to activation of mechanical modes of a structure
Electricity may be produced by coupling with a circuit (electromagnetic induction)
Figure: Elastic magnet in a fluid flow coupled with coil.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
2 / 37
Works at UME
Piezoelectricity
Magnetic Dipoles
[J. Lee, J. Boisson, C. Rouby]
[O. Doar´
e, S. Michelin (LadHyx)]
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
3 / 37
Works at LMS
Magnetised Elastomer
[L. Bodelot]
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
4 / 37
Many people are working on this challenging
problem!
Me:
Modelling the equations of motion of a
magnetic patch
Postdoc: Coup de Pouce F2M 2013, encadr´
e
par Corinne Rouby et Laurence Bodelot
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
5 / 37
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
6 / 37
Continuum Patch: Complex Problem
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
7 / 37
Continuum Patch: Complex Problem
Energy Of A Magnetic Patch
Magnetisation Field
in the deformed configuration M is not
uniform
(a)
calculating the total magnetic energy
for a deformed configuration is complex
M
New idea (after a few months)
work with chain of rectangular magnets
take a limit of infinite degrees of
freedom at the end
(b)
M
On fait par morceaux!
Figure: (a): Undeformed configuration. (b):
Deformed configuration.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
8 / 37
Discrete Rectangular Magnets
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
9 / 37
Discrete Rectangular Magnets
Rectangular Magnet With Uniform Magnetisation
Specifications
dimensions: 2Lx , 2Ly , 2h
ˆ
magnetisation : M0 z
volume specified by the vector r0
z
r
To do
2Lx
2Ly
2h
x
O
r0
1
Determine the magnetic field created by
the magnet at a location r
2
Determine the energy of interaction
between two magnets
M = M0zˆ
y
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
10 / 37
Discrete Rectangular Magnets
Rectangular Magnet With Uniform Magnetisation
Tricks
1
Absence of macroscopic currents allows the
definition of a scalar potential Φ(r).
Magnetic field: H(r) = −∇Φ
2
Uniform magnetisation allows clever
manipulations of the divergence theorem
In the end the scalar potential for the magnet
on the left reads
"I
#
I
0
0
dSup
dSlow
M0
−
Φ(r) =
0
0
4π
|r − r0 |
|r − r0 |
Sup
Slow
M = M0zˆ
Figure: Uniformly magnetised rectangular
magnet.
Surface integrals! Upper and lower surfaces:
sheets of ”magnetic charge”.
This looks simple ... But what happens when
the magnets is rotated and displaced from the
origin?
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
11 / 37
Discrete Rectangular Magnets
Energy of Interaction
Two interacting magnets
z
O
x
θj
rup
j
θi
rup
i
rlow
i
Oi
µ0
2
I
Sj
µ0
Φi (Mj · nj ) dSj +
2
rlow
j
Oj
Etot ≡ Eij + Eji =
I
Φj (Mi · ni ) dSi
Si
Explicit form of such expression is impossible
analytically
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
12 / 37
Discrete Rectangular Magnets
Energy of Interaction
So far we have seen
Mutual energy is expressed as a double
surface integral! (difficult)
A solution for the potential is needed
Therefore: we need to approximate the
energy integrals!
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
13 / 37
Calculating the Energy
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
14 / 37
Calculating the Energy
Multipole Expansion
Approximation 1: Multipole Expansion
Expansion formula
Given the Legendre polynomials Pn one may write
a multipole expansion for the potential
Φ(r) =
I
∞
1 X 1
r · r0
0 n
(r
)
P
M · n0 dS 0
n
4π n=0 rn+1 S 0
r · r0
Expansion depends on reference frame
Accuracy increases with n
For spherical geometry, n = 2 (dipole) is exact
solution, but what about the present
rectangular geometry?
Figure: Angular plots of poles
(figurative example).
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
15 / 37
Calculating the Energy
Multipole Expansion
Approximation 1: Multipole Expansion
Calculation by hand
Expansion up to n = 5
4π
Φ(r) =
M0


L2
8hLx Ly z
12hLx Ly z
L2
20hLx Ly z 2 2
y
2 2
2 2
2
 x +
x Lx + y Ly + z h
−
+h +
r3
r5
3
3
r7
+
Lx Ly hz 4
2
2
2 2
4
2 2
4
9Lx + 10Lx Ly + 30Lx h + 9Ly + 30Ly h + 45h
3r 7
−
+
14Lx Ly hz 4 2
2
2 2
2
2 2
2 2 2
9Lx x + 5Lx Ly x + 5Lx Ly y + 15Lx h x
3r 9
2 2 2
4 2
2 2 2
2 2 2
4 2
+5Lx h z + 9Ly y + 15Ly h y + 5Ly h z + 15h z
21Lx Ly hz 4 4
2
2 2 2
2 2 2 2
4 4
2 2 2 2
4 4
3Lx x + 10Lx Ly x y + 10Lx h x z + 3Ly y + 10Ly h y z + 3h z
+ ...
r 11
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
16 / 37
Calculating the Energy
Multipole Expansion
Multipole Expansion Test: close magnets
(a)
Figure: Two magnets, close configuration.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
17 / 37
Calculating the Energy
Multipole Expansion
Multipole Expansion Test: close magnets
4π/M0 Φ
·10−3
1
0.5
[m]
2
3
4
·10−2
(a)
(b)
Figure: Two magnets, close configuration.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
18 / 37
Calculating the Energy
Multipole Expansion
Multipole Expansion Test: close magnets
4π/M0 Φ
·10−3
1
0.5
[m]
2
3
4
·10−2
(a)
(b)
(c)
Figure: Two magnets, close configuration.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
19 / 37
Calculating the Energy
Multipole Expansion
Multipole Expansion Test: far magnets
4π/M0 Φ
·10−4
3
2
1
[m]
3
4
·10−2
(a)
(b)
(c)
Figure: Two magnets, far configuration.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
20 / 37
Calculating the Energy
Multipole Expansion
So far we have seen
Multipole expansion calculated up to
n=5
When magnets are close results are
quite bad!
Therefore: integrals need to be calculated
otherwise!
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
21 / 37
Calculating the Energy
Simpson 2D Rule
Approximation 2: Simpson’s Rule
Function approximation
Simpson’s 2D rule
The integral
I
I
Eij =
Si
Sj
dSi dSj
|rj − ri |
can be approximated by
"
I = Simpson2D


#
1

Simpson2D  −
r
rj
∆yi ,∆yj
i ∆x ,∆x
i
j
Figure: Bar approximation of continuous
2D function.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
22 / 37
Calculating the Energy
Simpson 2D Rule
Algorithm For Simpson 2D
|
(xj,− , xi,− )
•
•
•
•
•
···
···
.
.
.
•
(xj,+ , xi,− )
.
.
.
•
•
.
.
.
•
•
.
•
•
 1
 4
 2


 .
 .

 .

 2

4
1
|
(xj,− , xi,+ )
•
.
.
.
.
.
.
.
···
•
•
···
•
(xj,+ , xi,+ )
{z
}
Discrete input grid for integrand evaluation
.
≡ FI
Given
4
16
8
2
8
4
.
.
.
8
16
4
.
.
.
4
8
2
···
···
···
2
8
4
.
.
.
.
.
.
···
4
···
8
···
2
{z
Matrix of weights
4
16
8
.
.
.
8
16
4
1
4
2





. 
. 

. 

2 

4
1
}
≡ W2D
eT
1, ..., 1)
N = (1,
| {z }
N times
◦ = Hadamard product

1


Simpson2D
up
up rj − ri 
=
∆xj ∆xi T
eNx (W2D ◦ FI ) eNxi
j
9
∆xi ,∆xj
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
23 / 37
Calculating the Energy
Simpson 2D Rule
Algorithm For Simpson 2D
For this method to be useful
|
(xj,− , xi,− )
•
•
•
•
•
···
···
•
•
(xj,− , xi,+ )
•
1
.
.
.
•
(xj,+ , xi,− )
.
.
.
•
•
.
.
.
•
•
.
.
.
.
•
•
.
.
.
•
(xj,+ , xi,+ )
Matrix FI must be filled up
symbolically not numerically!
2
Algorithm must return a symbolic
expression for Eij , even though the
solution is approximated
.
.
···
···
{z
Discrete input grid for integrand evaluation
}
Software to accomplish this
≡ FI
Matlab
Mathematica
Maxima
....
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
24 / 37
Calculating the Energy
Simpson 2D Rule
Simpson 2D Test
Plots
Numeric Test
f (x, y) = sin(πx) sin(πy);
(x, y) ∈ [0, 1] × [0, 1]
R
1. S f (x, y) dx dy = 4/π 2 ≈ 0.4053
2. Simpson2D [f (x, y)]∆x,∆y = 0.4071
error ≈ 0.5% with very few domain subdivisions!
Simpson 2D very accurate method for double
integrals
Figure: Top: f (x, y) = sin(πx) sin(πy).
Bottom: Simpson 2D mesh with 4×4
subdivisions.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
25 / 37
Calculating the Energy
Simpson 2D Rule
So far we have seen
Eij can be calculated semi-analytically
using symbolic Simpson 2D
Ready to set up Lagrange equations!
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
26 / 37
Lagrange Equations
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
27 / 37
Lagrange Equations
Lagrange Equations
Clamped-Free Chain
M
zˆ
O x
ˆ
Constrained Equations
g
M
(xO1 , zO1 )
−θ2
(xO2 , zO2 )
M
(xO3 , zO3 )
θ3
Degrees of freedom = number of
magnets
Angles θi most natural choice for
describing the system
BUT Eij is written in terms of
xO i , z O i , θ i
L=T −V
N X
b 2
I 2
2
T =
x˙ O + z˙O + θ˙i
i
i
2
2
i=1


N
N
X
X

V =
Eij − b xOi g 
∂L
d ∂L
−
=
dt ∂ ˙ i
∂i
i=1
j=1
NC
Xz
Cx
X
∂gk
∂fk
λk +
µk
∂i
∂i
k=1
k=1
N
T
= [θ, xO , zO ]
Constraint equations

fi = xOi
− L 2
i−1
X

cos θj + cos θi  = 0
j=1

gi = zOi − L 2
i−1
X

sin θj + sin θi  = 0
j=1
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
28 / 37
Lagrange Equations
Linearised Equations
Quadratic form for energy
Linearised Equations
N
X
∂Eij Eij = Eij +
m +
∂m 0
0
m=1
1
2
N X
N
X
m=1 n=1
¨ + Kθ = 0
Bθ
Eigenvalue Equation!
2
∂ Eij m n +
∂ m ∂n 0
N X
N X
N
X
O(p q r )
p=1 q=1 r=1
We can do this because Eij is known
semi-analytically
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
29 / 37
Lagrange Equations
Eigenmodes
Eigenmodes
Plots
Figure: Eigenmodes for a chain of N = 5
magnets. First mode at zero frequency not
shown.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
30 / 37
Preliminary Validation of Model
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
31 / 37
Preliminary Validation of Model
Experimental Setup
(a)
(b)
(c)
Figure: (a) Rectangular magnets without cover. (b) Rectangular magnets with cover (top view).
(c) Rectangular magnets with cover (flat view)
(d)
(e)
Figure: (d) High Speed Camera. (e) Tracking framework in Matlab.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
32 / 37
Preliminary Validation of Model
Experimental Results (preliminary)
Two chains with different separation distances: bc, sc
Two configurations: with gravity / without gravity
bc2
sc2
bc2
sc2
model (Hz)
data (Hz)
3.6
4.5
1.9
3.2
3.4
4.2
1.9
2.9
+g
+g
no g
no g
More experiments at ENSTA right now ....
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
33 / 37
Conclusion and Perspectives
Contents
1
Continuum Patch: Complex Problem
2
Discrete Rectangular Magnets
Energy of Interaction
3
Calculating the Energy
Multipole Expansion
Simpson 2D Rule
4
Lagrange Equations
Eigenmodes
5
Preliminary Validation of Model
6
Conclusion and Perspectives
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
34 / 37
Conclusion and Perspectives
Conclusions
Chain of magnets: solved!
However, many more results are needed! ....
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
35 / 37
Conclusion and Perspectives
Perspectives
Elastic Magnetic Flag
Limit of infinite DoF
Figure: System composed of: elastic magnetic
flag (produced at LMS) and coils
Figure: Eigenfunctions of chain of magnets in
the limit of infinite degrees of freedom.
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
36 / 37
Conclusion and Perspectives
Perspectives
Passage discrete/continuum could give useful results
System need to be coupled with: coil AND fluid
Nonlinearities need to be addressed (so far model is linear)
M. Ducceschi
Dynamique d’une chaˆ
ıne d’aimants
12 - 05 - 2015
37 / 37