Page 399 - Fundamentals of Magnetic Thermonuclear Reactor Design

Mechanics of Magnetic Fusion Reactors Chapter | 12 377

H

( +
−z i ϕ )
2 π h 2 − 2 ( R 2 i + R 0 2 ) cos ϕ + R R 3cos 2


i 0
⋅ ∫ ∫ 5/2 ddh,
ϕ
H 0 ( R 2 + R 0 2 − RR2 cos ϕ + h 2 )
− −z i i i 0 2
2 ⋅∫−H2−ziH2−zi∫0πh −2Ri2+R0
2
2cos+RiR 3+cos Ri2+R02−2Ri
where =−h z z . i h=z−zi 0 2
The corresponding potential of electromagnetic forces R cos+h 5/2ddh,
0
1
cs
2
U () () =u c ( CS) ⋅u .
i
i
2 i i Ui(cs)ui=12ci(CS)⋅ui2.
The potential of interaction of all TFCs and the CS
1 N PF
m
U () (uu, ,..., u ) = c ( CS) 2
u ,
2
CS
i
2 ∑ i i UCS(m)ui,u ,...,uNPF=12∑i=1NP
N PF
= i 1 2
Fci(CS)ui2,
or in standard quadratic form
() 1 N PF N PF ( ) 1 ()
m
CS
m
U = ∑ ∑ a u u = u Τ ⋅Α ⋅u,
CS
2 = i 1 = j 1 ij ij 2 CS
( ) ( )
CS
CS
a ij = δ c . UCSm=12∑i=1NPF∑j=1NPFaijCSuiuj=
ij i
Next, we determine the elastic forces. Each PFC has N elastic mechanical 12uT⋅ACSm⋅u,aijCS=δijciCS.
SP
(r)
supports. The stiffness of each support in the radial and toroidal directions is c
(t)
(r)
(t)
and c , respectively. The elastic support are oriented such that c /c << 1. For
the i-th PFC, the supports’ total stiffness relative to displacement along axis х is
N SP  2 π   2 π 
t ()
c i e () = c i r () ∑ cos 2  j  + c sin 2  j .

i
2
= j 1  N SP   N SP  ci(e)=ci(r)∑j=1NSPcos 2πNSPj+ci
2
(t)sin 2πNSPj.
The elastic forces’ overall potential
1 N
e () 2
U e () (uu, ,..., u ) = c u ,
2
1
SP
N PF
2 ∑ i i USP(e)u ,u ,...,uNPF=12∑i=1Nc
= i 1 1 2
i(e)ui2,
or, in standard form
∑
∑
e ()
U e () = 1 N PF N PF au u = 1 u Τ ⋅Α e () ⋅u,
ij
SP
2 = i 1 = j 1 ij 2 SP
e ()
a ij e () = δ c . USP(e)=12∑i=1NPF
ij ij
∑j=1NPFaij(e)uiuj=-
The overall potential of all forces acting on the PFC system is the sum total
of the obtained potentials of magnetic and elastic forces, that is 12uT⋅ASP(e)⋅u,aij(e)=δijcij(e).
m
m
()
U (uu , 1 2 ,..., N PF ) = U PF +U () +U SP e () ,
u
CS
1
m
e ()
m
U = u T ⋅ ⋅ , u A = Α () + Α () + Α .
A
2 PF CS SP Uu ,u ,...,uNPF=UPF(m)+UCS(m
1
2
)+USP(e),U=12uT⋅A⋅u, A=APF(
m)+ACS(m)+ASP(e).