Inertia tensors for non-typical rigid bodies

Question

Calculate inertia tensors

This topic inspired me to experiment with calculating tensors of more complex shapes of rigid bodies (I did not find them in the Mathematica database).

For simple shapes of rigid bodies, everything works great:

MomentOfInertia[Ball[{0, 0, 0}, R]];

MomentOfInertia[Cone[{{0, 0, 0}, {0, a, 0}}, R]];

MomentOfInertia[Cylinder[{{0, -1/2, 0}, {0, 1/2, 0}}, R]];

Remark: https://mathematica.stackexchange.com/a/62895/67019 The code from here also works and gives similar results.

My question: And how to calculate the inertia tensor for more complex rigid bodies. For example, for a sector of a torus or a ring with a rectangular cross section?

This picture from SolidWorks.

Answer 1

Here is an example for half a torus.

r0 = 1; (*center radius*)
r1 = 0.2;(*outer radius*)
Region[ParametricRegion[{(r0 + r*Cos[θ]) Cos[ϕ], (r0 + r*Cos[θ]) Sin[ϕ], 
   r*Sin[θ]}, {{r, 0, r1}, {θ, 0, 2 π}, {ϕ, 
    0, π}}], Axes -> True]

The mass density is assumed to be 1. For other values you simply multiply the result by this value. The inertia tensor relative to the coordinate axes is then:

reg = ParametricRegion[{(r0 + r*Cos[θ]) Cos[ϕ], (r0 + 
       r*Cos[θ]) Sin[ϕ], r*Sin[θ]}, {{r, 0, r1}, {θ, 0, 2 π}, {ϕ,0, π}}];
tensor = Outer[Times, {x, y, z}, {x, y, z}]
NIntegrate[tensor, {x, y, z} ∈ reg]

(* {{x^2, x y, x z}, {x y, y^2, y z}, {x z, y z, z^2}} *)

(* {{0.20109, 0.00010941, 2.40669*10^-6}, {0.0000312697, 0.200579, 
  0.0000373482}, {-0.0000400759, -9.15108*10^-7, 0.00384308}} *)

Addendum

The cross section is given by the terms: r*Cos[θ]and r*Sin[θ]. To change the cross section we simply need to change these terms. E.g. a quadratic cross section:

r0 = 1; 
w = 0.2;
Region[ParametricRegion[{(r0 + x1) Cos[ϕ], (r0 + x1) Sin[ϕ],
    y1}, {{x1, -w, w}, {y1, -w, w}, {ϕ, 0, π}}], Axes -> True]

Answer 2

• According to the document of MomentOfInertia, the tensor should be

{{y^2 + z^2, -x*y, -x*z}, {-y*x, x^2 + z^2, -y*z}, {-z*x, -z*y, 
  x^2 + y^2}}

result1=Integrate[{{y^2 + z^2, -x*y, -x*z}, {-y*x, 
   x^2 + z^2, -y*z}, {-z*x, -z*y, x^2 + y^2}}, {x, y, z} ∈ 
  Ball[{0, 0, 0}, R]];
result2=MomentOfInertia[Ball[{0, 0, 0}, R]];
result1 == result2

True.

• For general region,

Clear[reg];
reg = RegionDifference[
   RegionProduct[Annulus[{0, 0}, {1, 2}], Line[{{0}, {1}}]], 
   Cuboid[{0, 0, 0}, {2, 2, 2}]];
Region[reg, ViewPoint -> {1, 1, 1}]
MomentOfInertia[DiscretizeRegion@reg, {0, 0, 0}]
Integrate[{{y^2 + z^2, -x*y, -x*z}, {-y*x, 
   x^2 + z^2, -y*z}, {-z*x, -z*y, x^2 + y^2}}, {x, y, z} ∈ 
  DiscretizeRegion@reg]