Download - Mdd Paper Cone Equations
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7/25/2019 Mdd Paper Cone Equations
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x i=x i(1, 2 , 3)
y i=y i(1 ,2, 3 )
gij= xr
i
xr
j
gij=
i
xr
j
xr
Gij= y
r
i
yr
j
Gij=
i
yr
j
yr
I1=grs
Grs
I2=grs
Grs
I3=grs
Grs
g=|g ij|
G=|Gij|
I3=1
W=W(I1, I2)
ij= gij+Bij+Gij
=2 W/ I1
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7/25/2019 Mdd Paper Cone Equations
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=2 W/ I2
Bij=(gij grsg ir gjs)Grs
ij i+ Fj= fj
ij
ni=pj
ij=(Gjj /Gii)1 /2 ij
x1=f(r)cos
x2=f(r) sin
x3=z /
y1=rcos
y2=rsin
y3=z
f
( '(r ))2
0 00 ( f(r ))2
0
g ij= [02 ]
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7/25/2019 Mdd Paper Cone Equations
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gij=[(f'(r ))2 0 0
0 ( f(r ))2 00 0 1 /2]
Gij=[
1 0 0
0 r2
0
0 0 1]Gij=[
1 0 0
0 r2
0
0 0 1]
gij=
[1/ 0 00 1/ r2 0
0 0 2]
gij=[ 0 0
0 r2
0
0 0 2]
x1= rcos
x2= rsin
x3=z /
I1=2+2/
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7/25/2019 Mdd Paper Cone Equations
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I2=1/2+2
I3=1
Bij=[
+1 /2 0 00 1/2 r2+/r2 00 0 2
] p
r
=r 2
p
=
p
z=0
ij r=
ij,r+rm
i rmj
ijr=grsijs
ijs= 2x
r
i
j
xr
s
p=0.5 r2 2
+ B
11=r 22=
+ (+1 /
2 ) 12
r2
2+B
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7/25/2019 Mdd Paper Cone Equations
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33=2+2
1
2r
2
2+B
23=31=12=0
n1=cos
1, n
20,n
1=sin
1
11=!1 , 33 sin 1=!
3"# r=r1 on c$r%& s$rf"c&
!1=0 , r=r1
B=1
2
2r
1
21
(+1 /2 )
11=22=11=r2 22
=
r
(12r2)1
2
2
r
(12r2)+( 21 )+( 12 )
33=1
2
2
33=
1
2
2("2
r2)+(21)+(12 )
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7/25/2019 Mdd Paper Cone Equations
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0
2(
0
" /
33
r''r=0
(
21
)+
(
1
2
)+
2
"2
4 =0
!0
3=
2
"2
4
!=2 ( r1!
3= 1
2(( " 22 sin
1) r
1)
W=*1(I13 )+*2(I23)
4++ 3+(n21 )+=0
r
(12r2)
11=22=1
2
2
33=
1
4
2[(2 r12"2
1)2r2]
33=
1
4
2 [( 2 r12"2 )2 r2 ]
12=
23=
31=0