- Source: Sphericity
Sphericity is a measure of how closely the shape of an object resembles that of a perfect sphere. For example, the sphericity of the balls inside a ball bearing determines the quality of the bearing, such as the load it can bear or the speed at which it can turn without failing. Sphericity is a specific example of a compactness measure of a shape.
Sphericity applies in three dimensions; its analogue in two dimensions, such as the cross sectional circles along a cylindrical object such as a shaft, is called roundness.
Definition
Defined by Wadell in 1935, the sphericity,
Ψ
{\displaystyle \Psi }
, of an object is the ratio of the surface area of a sphere with the same volume to the object's surface area:
Ψ
=
π
1
3
(
6
V
p
)
2
3
A
p
{\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}}
where
V
p
{\displaystyle V_{p}}
is volume of the object and
A
p
{\displaystyle A_{p}}
is the surface area. The sphericity of a sphere is unity by definition and, by the isoperimetric inequality, any shape which is not a sphere will have sphericity less than 1.
Ellipsoidal objects
The sphericity,
Ψ
{\displaystyle \Psi }
, of an oblate spheroid (similar to the shape of the planet Earth) is:
Ψ
=
π
1
3
(
6
V
p
)
2
3
A
p
=
2
a
b
2
3
a
+
b
2
a
2
−
b
2
ln
(
a
+
a
2
−
b
2
b
)
,
{\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}},}
where a and b are the semi-major and semi-minor axes respectively.
Derivation
Hakon Wadell defined sphericity as the surface area of a sphere of the same volume as the particle divided by the actual surface area of the particle.
First we need to write surface area of the sphere,
A
s
{\displaystyle A_{s}}
in terms of the volume of the object being measured,
V
p
{\displaystyle V_{p}}
A
s
3
=
(
4
π
r
2
)
3
=
4
3
π
3
r
6
=
4
π
(
4
2
π
2
r
6
)
=
4
π
⋅
3
2
(
4
2
π
2
3
2
r
6
)
=
36
π
(
4
π
3
r
3
)
2
=
36
π
V
p
2
{\displaystyle A_{s}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{p}^{2}}
therefore
A
s
=
(
36
π
V
p
2
)
1
3
=
36
1
3
π
1
3
V
p
2
3
=
6
2
3
π
1
3
V
p
2
3
=
π
1
3
(
6
V
p
)
2
3
{\displaystyle A_{s}=\left(36\,\pi V_{p}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{p}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}
hence we define
Ψ
{\displaystyle \Psi }
as:
Ψ
=
A
s
A
p
=
π
1
3
(
6
V
p
)
2
3
A
p
{\displaystyle \Psi ={\frac {A_{s}}{A_{p}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{p}\right)^{\frac {2}{3}}}{A_{p}}}}
Sphericity of common objects
See also
Equivalent spherical diameter
Flattening
Isoperimetric ratio
Rounding (sediment)
Roundness
Willmore energy
References
External links
Grain Morphology: Roundness, Surface Features, and Sphericity of Grains
Kata Kunci Pencarian:
- Batuan sedimen
- Karakteristik teknik bahan pertanian
- Sphericity
- Mauchly's sphericity test
- Spheric
- Spherical coordinate system
- Spherical Earth
- Spherical harmonics
- Spherical aberration
- Spherical cow
- Spherical trigonometry
- Greenhouse–Geisser correction
