Problem 11.35
Given:
A disk of inner radius
a
and outer radius
b
is subjected to an angular velocity
ω
.
The disk is
constrained at
r
=
a
, so that the radial displacement
u
is zero.
At
r
b
, the disk is free of applied
forces.
Derive formulas for the constants of integration
C
1
and
C
2
as functions of
ω
, the material
properties and radii.
Solution:
The boundary conditions are:
at
ra
=
u0
=
at
rb
=
σ
rr
0
=
Eq. 11.47 with T=0
u
1
ν
2
−
()
−
8E
⋅
ρ
⋅
r
3
⋅
ω
2
⋅
C
1
r
⋅
+
C
2
r
+
=
at
=
1
ν
2
−
⋅
ρ
⋅
a
3
⋅
ω
2
⋅
C
1
a
⋅
C
2
a
+
=
or
C
2
1
ν
2
−
⋅
ρ
⋅
a
4
⋅
ω
2
⋅
C
1
a
2
⋅
−
=
Eq. 11.49
σ
rr
E
1
ν
2
−
3
ν
+
−
1
ν
2
−
⋅
⋅
ρ
⋅
r
2
⋅
ω
2
⋅
1
ν
+
C
1
⋅
+
1
ν
−
r
2
C
2
⋅
−
⎡
⎢
⎣
⎤
⎥
⎦
⋅
=
at
=
σ
rr
0
=
1
ν
+
C
1
⋅
1
ν
−
b
2
C
2
⋅
−
3
ν
+
1
ν
2
−
⋅
⋅
ρ
⋅
b
2
⋅
ω
2
⋅
=
Eliminate
C
2
:
1
ν
+
C
1
⋅
1
ν
−
b
2
1
ν
2
−
⋅
ρ
⋅
a
4
⋅
ω
2
⋅
C
1
a
2
⋅
−
⎛
⎜
⎝
⎞
⎟
⎠
⋅
−
3
ν
+
1
ν
2
−
⋅
⋅
ρ
⋅
b
2
⋅
ω
2
⋅
=
C
1
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 Spring '10
 MechanicsofSolids
 Strain

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