Constant stress solid disk rotor of flywheel for flywheel energy storage system and design method thereof
A constant stress solid disk rotor of a flywheel has an outer shape having a plane-symmetric upper surface and lower surface, an outer circumferential radius b, and a rotation center thickness h 0 , and includes a thickness decreasing region which decreases monotonously in thickness from a rotation center to a connection radius a and a constant thickness region located on an outer edge of the thickness decreasing region and having a constant thickness h a from the connection radius a to the outer circumferential radius b. Shape parameters including the outer circumferential radius b, the rotation center thickness h 0 , the connection radius a, and the outer edge thickness h a satisfy an equation below. Here, ν is a Poisson's ratio of a rotor material. a b = 1 2 ( - 2 1 - v ( 1 + v - 2 ln ( h a h 0 ) ) + ( 2 1 - v ( 1 + v - 2 ln ( h a h 0 ) ) ) 2 + 4 ( 3 + v ) 1 - v )
1. A constant stress solid disk rotor of a flywheel, having an outer shape having an upper surface and a lower surface which are plane-symmetric with respect to a single-center rotation plane perpendicular to a rotation axis, an outer circumferential radius b, and a rotation center thickness h 0 , the constant stress solid disk rotor of the flywheel having a shape including a thickness decreasing region which decreases monotonously in thickness from a rotation center to a connection radius a and a constant thickness region located on an outer edge of the thickness decreasing region and having a constant thickness h a from the connection radius a to the outer circumferential radius b, wherein
shape parameters including the outer circumferential radius b, the rotation center thickness h 0 , the connection radius a, and the outer edge thickness h a satisfy an equation below:
[
Expression
1
]
a
b
=
1
2
(
-
2
1
-
v
(
1
+
v
-
2
ln
(
h
a
h
0
)
)
+
(
2
1
-
v
(
1
+
v
-
2
ln
(
h
a
h
0
)
)
)
2
+
4
(
3
+
v
)
1
-
v
)
)
where ν is a Poisson's ratio of a rotor material.
2. The constant stress solid disk rotor of the flywheel according to claim 1 , wherein the connection radius a and the outer edge thickness h a are set without depending on a rotation angular velocity.
3. The constant stress solid disk rotor of the flywheel according to claim 1 , wherein the thickness decreasing region is formed in a shape in which an in-plane stress of the thickness decreasing region is always invariant entirely in the thickness decreasing region.
4. The constant stress solid disk rotor of the flywheel according to claim 3 , wherein the constant thickness region is formed in a shape in which an in-plane stress of the constant thickness region decreases monotonously from a stress value which is invariant in a plane of the thickness decreasing region toward the outer circumferential radius b from the rotation center.
5. The constant stress solid disk rotor of the flywheel according to claim 1 , wherein a thickness h of the thickness decreasing region is expressed by an expression below:
[
Expression
2
]
h
(
r
)
=
h
o
e
ln
(
h
a
h
0
)
a
2
r
2
.
6. The constant stress solid disk rotor of the flywheel according to claim 1 , wherein when rotating with the thickness decreasing region producing an in-plane stress σ a , a rotation angular velocity ω is expressed by an expression below:
[
Expression
3
]
ω
=
-
σ
a
ln
(
h
a
h
0
)
ρ
a
2
where ρ is a density of the rotor material.
7. The constant stress solid disk rotor of the flywheel according to claim 1 , wherein assuming that a yield strength of the rotor material is σ y , a limit energy density D FR is expressed by an equation below:
[
Expression
4
]
D
FR
=
-
(
(
a
b
)
4
(
h
a
h
0
)
ln
(
h
a
h
0
)
-
(
h
a
h
0
-
1
)
)
+
1
2
(
h
a
h
0
)
(
ln
(
h
a
h
0
)
)
2
(
1
-
(
a
b
)
4
)
)
(
a
b
)
2
(
(
a
b
)
2
(
h
a
h
0
-
1
)
+
(
h
a
h
0
)
ln
(
h
a
h
0
)
(
1
-
(
a
b
)
2
)
)
×
σ
y
ρ
.
8. A method for designing the constant stress solid disk rotor of the flywheel according to claim 7 , wherein any three parameters among six parameters of the limit energy density D FR , a mass, the outer circumferential radius b, the rotation center thickness h 0 , the connection radius a, and the outer edge thickness h a of the constant stress solid disk rotor of the flywheel are given, and remaining three parameters are determined.
9. A method for designing the constant stress solid disk rotor of the flywheel according to claim 1 , wherein any three parameters among four parameters of the outer circumferential radius b, the rotation center thickness h 0 , the connection radius a, and the outer edge thickness h a are given, and a remaining one parameter is determined.