A I = [ I x x − I x y − I x z − I x y I y y I y z − I x z − I y z I z z ] A_{I}=\left[\begin{array}{ccc}{I_{x x}} & {-I_{x y}} & {-I_{x z}} \\ {-I_{x y}} & {I_{y y}} & {I_{y z}} \\ {-I_{x z}} & {-I_{y z}} & {I_{z z}}\end{array}\right] AI=⎣⎡Ixx−Ixy−Ixz−IxyIyy−Iyz−IxzIyzIzz⎦⎤
w h e r e where where, I x x = ∭ V ( y 2 + z 2 ) ρ d v I y y = ∭ V ( x 2 + z 2 ) ρ d v I z z = ∭ V ( x 2 + y 2 ) ρ d v I x y = ∬ V x y ρ d v I z z = ∬ V x z ρ d v I y z = ∭ V y z ρ d v \begin{aligned} I_{x x} &=\iiint_{V}\left(y^{2}+z^{2}\right) \rho d v \\ I_{y y} &=\iiint_{V}\left(x^{2}+z^{2}\right) \rho d v \\ I_{z z} &=\iiint_{V}\left(x^{2}+y^{2}\right) \rho d v \\ I_{x y} &=\iint_{V} x y \rho d v \\ I_{z z} &=\iint_{V} x z \rho d v \\ I_{y z} &=\iiint_{V} y z \rho d v \end{aligned} IxxIyyIzzIxyIzzIyz=∭V(y2+z2)ρdv=∭V(x2+z2)ρdv=∭V(x2+y2)ρdv=∬Vxyρdv=∬Vxzρdv=∭Vyzρdv
其中 I x x , I y y , I z z I_{x x}, I_{y y}, I_{z z} Ixx,Iyy,Izz称为惯量距,其余三个交叉项称为惯量积。对于一个刚体来说,这六个相互独立的量取决于所在坐标系的位姿。