MDD Paper Cone Equations
MDD Paper Cone Equations
MDD Paper Cone Equations
y i= y i ( 1 , 2 , 3 )
gij =
ij
g=
xr xr
i j
i j
xr xr
r
Gij =
y y
i
j
i j
G = r
r
y y
ij
rs
I 1 =g Grs
I 2 =g rs Grs
I 3 =grs Grs
g=|g ij|
G=|Gij|
I 3 =1
W =W ( I 1 , I 2)
ij
ij
ij
= g + B + G
=2 W / I 1
ij
=2 W / I 2
B ij =( gij g rs g ir g js )Grs
ij i + F j= f j
ij ni= p j
ii 1 /2 ij
ij =(G jj /G )
0
0
0 (f (r ))2
ij
g = [ 0 2 ]
(f ' (r ))2
0
0
2
gij =
0
( f (r ))
0
0
0
1 / 2
[ ]
[ ]
[
]
1 0 0
G = 0 r2 0
0 0 1
ij
1 0 0
G ij = 0 r 2 0
0 0 1
1/
0
0
gij = 0 1/ r 2 0
0
0
2
gij = 0
0
0
0
2
r
0
2
0
x 1= rcos
x 2= rsin
x 3=z /
I 1 = +2/
I 2 =1/ 2 +2
I 3 =1
+1 / 2
0
0
2 2
2
B =
0
1/ r + /r
0
0
0
2
ij
p
=r 2
r
p p
=
=0
z
j
ij r = ij ,r + irm rm
rs
ij =g ijs
2 x r x r
i
j
s
ijs =
p=0.5 r
11 =r 22= +
+B
( +1 / 2 ) 1 r 2 2 +B
2
1
33= 2 +2 r 2 2 + B
2
23
31
12
= = =0
n1=cos 1 , n2 0,n 1=sin 1
11
33
P =0 , r=r 1
1
1
2 2
2
B= r 1 ( +1 / )
2
11 = 22= 11=r 2 22
r
( 12r 2 )
1
2
2
( 12r 2 )+ 2
1
1
+( 2 )
1
33 = 2
2
)(
) (
1
a
1
1
33 = 2 r 2 + 2 + 2
2
2 a/
33 rddr=0
) (
1
1 a
+ 2 +
=0
2 a2
P=
4
3
0
P=2 r 1 P =
1
2 2
( a sin 1 . r 1)
2
W =C1 ( I 13 ) +C 2( I 23)
4+ K 3 + ( n21 ) K =0
r
( 12r 2 )
1
11 = 22= 2
2
[(
1
a
33 = 2 2 r 21 2r 2
4
1
1
33 = 2 ( 2 r 21a2 ) 2 r 2
4
12= 23 = 31=0