Inertia tensor examples

Definitions, etc.

Off[General::spell1];

<< Graphics`Shapes`;

TF  =  TraditionalForm;

Example #1: Cylinder (uniform density)

Visualization

Show[Graphics3D[Cylinder[1, 3, 20]],  Boxed -> False,  
        ImageSize -> 100];

Volume of cylinder (radius = a, height = h)

V  =  π a^2 h;  V  //  TF

(Subscript[I, zz])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, zz]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))((x^2) + 
                    (y^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, zz])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz])  =  m (Subscript[I, zz])/(ρ V); 
(Subscript[I, zz])  //  TF

(Subscript[I, yy ])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yy]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))((x^2) + 
                    (z^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy])  =  m (Subscript[I, yy])/(ρ V)  //  Simplify  // 
          Expand; 
(Subscript[I, yy])  //  TF

(Subscript[I, xx ])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, xx]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))((y^2) + 
                    (z^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, xx])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, xx])  =  m (Subscript[I, xx])/(ρ V)  //  Simplify  //  
          Expand;
(Subscript[I, xx])  //  TF

Products of inertia

(Subscript[I, xy  ])

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, xy]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))(x y) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z

(Subscript[I, xz  ])

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, xz]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))(x z) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z

\(TraditionalForm\`\(\(I\_yz\)\_\ \ \)\)

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yz]) = (∫_(-h/2)^(h/2))(∫_0^a)(∫_0^(2  π))(y z) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z

Example #2: Thin, hollow cylinder

Visualization

Show[Graphics3D[Cylinder[1, 3, 20]],  Boxed -> False,  
        ImageSize -> 100];

Volume of hollow cylinder (radius = a, height = h, thickness = dr)

x  = r Cos[θ];  y  =  r Sin[θ]; 
V = (∫_0^h)(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) 
              r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
V  //  TF

(Subscript[I, zz])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, zz]) = (∫_(-h/2)^(h/2))(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) ((x^2) + 
                    (y^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, zz]) //  TF

Small dr approximation

(Subscript[I, zz])  =  
        (Subscript[I, zz])  /.  (dr^3)  -> 0;
(Subscript[I, zz])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz])  =  m (Subscript[I, zz])/(ρ V); 
(Subscript[I, zz])  //  TF

(Subscript[I, yy])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yy]) = (∫_(-h/2)^(h/2))(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) ((x^2) + 
                    (z^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, yy]) //  TF

Small dr approximation

(Subscript[I, yy])  =  
        (Subscript[I, yy])  /.  (dr^3)  -> 0;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy])  =  m (Subscript[I, yy])/(ρ V)  //  Simplify  //  
          Expand; 
(Subscript[I, yy])  //  TF

(Subscript[I, xx ])(same as (Subscript[I, yy ]))

Products of inertia

(Subscript[I, xy  ])

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, xy]) = (∫_(-h/2)^(h/2))(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) (x y) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, xy]) //  TF

(Subscript[I, xz  ])

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, xz]) = (∫_(-h/2)^(h/2))(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) (x z) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, xz]) //  TF

\(TraditionalForm\`\(\(I\_yz\)\_\ \ \)\)

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yz]) = (∫_(-h/2)^(h/2))(∫_(a - dr/2)^(a + dr/2))(∫_0^(2  π)) (y z) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, yz]) //  TF

Example #3: Cylinder with hollow center (uniform density)

Volume of cylinder (outer radius = b, inner radius = a, height = h)

V  =  π (b^2) h  -  π (a^2) h; 
V  //  TF

(Subscript[I, zz])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, zz]) = (∫_(-h/2)^(h/2))(∫_a^b)(∫_0^(2  π))((x^2) + 
                    (y^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, zz])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz])  =  m (Subscript[I, zz])/(ρ V)  //  Expand  //  
          Simplify; 
(Subscript[I, zz])  //  TF

(Subscript[I, yy ])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yy]) = (∫_(-h/2)^(h/2))(∫_a^b)(∫_0^(2  π))((x^2) + 
                    (z^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy])  =  m (Subscript[I, yy])/(ρ V)  //  Simplify  // 
            Expand  //  Together; 
(Subscript[I, yy])  //  TF

(Subscript[I, xx ])(same as (Subscript[I, yy ]))

Example #4: Thin disk

Volume of thin disk (radius = a, thickness = dh)

V =  dh π (a^2) ;
V  //  TF

(Subscript[I, zz])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, zz]) = (∫_(-dh/2)^(dh/2))(∫_0^a)(∫_0^(2  π)) ((x^2) + 
                    (y^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, zz]) //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz, disk])  =  
        m (Subscript[I, zz])/(ρ V); 
(Subscript[I, zz, disk])  //  TF

(Subscript[I, yy])

Computation in terms of density ρ

x  = r Cos[θ];  y  =  r Sin[θ]; 
(Subscript[I, yy]) = (∫_(-dh/2)^(dh/2))(∫_0^a)(∫_0^(2  π)) ((x^2) + 
                    (z^2)) ρ r \[DifferentialD]θ \[DifferentialD]r \[DifferentialD]z;
(Subscript[I, yy]) //  TF

Small dh approximation

(Subscript[I, yy])  =  
        (Subscript[I, yy])  /.  (dh^3)  -> 0;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy, disk])  =  m (Subscript[I, yy])/(ρ V)  //  
            Simplify  //  Expand; 
(Subscript[I, yy, disk])  //  TF

(Subscript[I, xx ])(same as (Subscript[I, yy]))

(Subscript[I, xx, disk])  =  (Subscript[I, yy, disk]);
(Subscript[I, xx, disk])  //  TF

Example #5: Cone

Visualization

Show[Graphics3D[Cone[1, 2, 20]], Boxed -> False,  
        ImageSize -> 100];

Volume of cone (base radius = b, height = h)

rule  =  a ->  b (h - z)/h;

dV  =  π (a^2)  /.  rule; 
dV  //  TF

V  =  (∫_0^h) 
            dV \[DifferentialD]z;
V  //  TF

Center of mass of cone (base radius = b, height = h)

(Subscript[cm, z])  =  (((∫_0^h) z dV \[DifferentialD]z)/((∫_0^h) dV \[DifferentialD]z));
(Subscript[cm, z])  //  TF

(Subscript[I, zz])

Center-of-mass relative differential volume element

rule  =  a ->  b (h - (z + (Subscript[cm, z])))/h;

dV  =  π (a^2)  /.  rule; 
dV  //  TF

Center-of-mass relative differential mass element

dm  =  ((Subscript[I, zz, disk]))/m (ρ dV)  /.  
          rule;
dm  //  TF

Computation in terms of density ρ

(Subscript[I, zz]) = (∫_(-h/4)^(3  h/4)) 
          dm \[DifferentialD]z;
(Subscript[I, zz])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz, cone]) = 
        m (Subscript[I, zz])/(ρ V)  //  
          Simplify;  
(Subscript[I, zz, cone]) //  TF

(Subscript[I, yy])

Center-of-mass relative differential mass element

dm  =  (((Subscript[I, yy, disk]))/
                  m (ρ dV)  +  ρ (z^2) dV) /.  
          rule;
dm  //  TF

Computation in terms of density ρ

(Subscript[I, yy]) = (∫_(-h/4)^(3  h/4)) 
          dm \[DifferentialD]z;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy, cone]) = 
        m (Subscript[I, yy])/(ρ V)  //  
          Simplify;  
(Subscript[I, yy, cone]) //  TF

(Subscript[I, xx ])(same as (Subscript[I, yy]))

Example #6:  Partial cone

Visualization

Show[Graphics3D[Cone[1, 2, 20]], Boxed -> False,  
        PlotRange -> {All,  All,  {-2,  .75}},  
        ImageSize -> 100];

Volume of cone (base radius = b, top radius = a, height = h)

rule  =  r ->  (b - a) (h - z)/h  +  a;

dV  =  π (r^2)  /.  rule; 
dV  //  TF

V  =  (∫_0^h) dV \[DifferentialD]z  //  
          Simplify;
V  //  TF

Center of mass of cone (base radius = b, height = h)

(Subscript[cm, z])  =  (((∫_0^h) z dV \[DifferentialD]z)/((∫_0^h) dV \[DifferentialD]z))  //  
          Simplify;
(Subscript[cm, z])  //  TF

(Subscript[I, zz])

Center-of-mass relative differential volume element

rule  =  (r ->  
              ExpandAll[((b - a) (h - (z + (Subscript[cm, z])))/h  + 
                    a)]) //  Simplify;
rule  //  TF

dV  =  π (r^2)  /.  rule; 
dV  //  TF

Center-of-mass relative differential mass element

dm  =  ((Subscript[I, zz, disk])  /.  a -> r)/
              m (ρ dV)  /.  rule;
dm  //  TF

Computation in terms of density ρ

(Subscript[I, zz]) = (∫_(-(Subscript[cm, z]))^(h - (Subscript[cm, z]))) 
          dm \[DifferentialD]z;
(Subscript[I, zz])  //  TF

(Subscript[I, zz])  =  (Subscript[I, zz])  //  Simplify;
(Subscript[I, zz])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, zz, cone]) = 
        m (Subscript[I, zz])/(ρ V)  //  
          FullSimplify;  
(Subscript[I, zz, cone]) //  TF

(Subscript[I, yy])

Center-of-mass relative differential mass element

dm  =  (((Subscript[I, yy, disk])  /.  a -> r)/
                  m (ρ dV)  +  ρ (z^2) dV) /.  
          rule;
dm  //  TF

Computation in terms of density ρ

(Subscript[I, yy]) = (∫_(-(Subscript[cm, z]))^(h - (Subscript[cm, z]))) 
          dm \[DifferentialD]z;
(Subscript[I, yy])  //  TF

(Subscript[I, yy])  =  (Subscript[I, yy])  //  Simplify;
(Subscript[I, yy])  //  TF

Conversion to mass representation (m = ρ V)

(Subscript[I, yy, cone]) = 
        m (Subscript[I, yy])/(ρ V)  //  
          FullSimplify;  
(Subscript[I, yy, cone]) //  TF

(Subscript[I, xx ])(same as (Subscript[I, yy]))