Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model

Adopt the model of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope: the reference mass distribution $\nu$, its total mass $m=\nu ({\mathbb{R}}^3)$ and its centred reference coordinates $\int \rho d\nu (\rho )=0$. Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, used only through Finiteness of the moments of a bounded finite Borel measure on ${\mathbb{R}}^3$ to make the polynomial moments of $\nu$ of degree at most two finite real numbers. Fix an inertial frame and a motion $x(t,\rho )=c(t)+R(t)\rho$ with configuration path $(c,R)$ of class $C^2$ (Configuration, configuration space and orientation of a rigid-material model in an inertial frame).

(a) Body inertia matrix. The body inertia matrix of the model is the inertia matrix of the reference mass distribution, $I:=I(\nu )=\int (\parallel \rho {\parallel }_2^2{I}_3-\rho {\rho }^{\mathsf{T}})d\nu (\rho )\in M_3(\mathbb{R}),$ with entries $I_{jk}=\int (\parallel \rho {\parallel }_2^2{\delta }_{jk}-{\rho }_j{\rho }_k)d\nu$ as in Mass distributions on ${\mathbb{R}}^3$: total mass, first moment and inertia matrix; the matrix is expressed in the standard basis of the reference coordinates. Its entries are finite real numbers, it is symmetric, and for every $\omega \in {\mathbb{R}}^3$, ${\omega }^{\mathsf{T}}I\omega =\int \parallel \omega \times \rho {\parallel }_2^{2}d\nu (\rho )\ge 0$ by Symmetry, the quadratic identity ${\omega }^{\mathsf{T}}I\omega =\int \parallel \omega \times \rho {\parallel }_2^{2}d\nu$, and the positive-definiteness criterion for the inertia matrix clauses 1-2; hence $I$ is symmetric and its quadratic form is positive semidefinite. The dimension is $[I]=\mu {\mathrm{\ell }}^2$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), read in kilogram metres squared. The model is nondegenerate when no line through the origin carries the whole mass, that is $\nu ({\mathbb{R}}^3\setminus \mathbb{R}\omega )>0$ for every $\omega \mathrm{\ne }0$, and degenerate otherwise; the connection between this criterion and the strict positivity and invertibility of $I$ is proved in the principal-axes theorem of this page.

(b) Spatial inertia tensor at the centre of mass. The spatial inertia tensor of the model at time $t$ is $I(t):=R(t)IR(t{)}^{\mathsf{T}}\in M_3(\mathbb{R}),$ the matrix of the same quadratic form in the frame's spatial coordinates: for every $n\in {\mathbb{R}}^3$, $n^{\mathsf{T}}I(t)n=(R(t{)}^{\mathsf{T}}n{)}^{\mathsf{T}}I(R(t{)}^{\mathsf{T}}n)$. Its entries are $C^2$ functions of $t$, since $R$ is of class $C^2$ and $I$ is constant; the matrix rotates with the body and returns to $I$ in the body-fixed frame, where $R=I_3$.

(c) Inertia tensor about a spatial point. For a spatial point $p\in {\mathbb{R}}^3$ let $a(t,p):=R(t{)}^{\mathsf{T}}(p-c(t))\in {\mathbb{R}}^3$ be the reference coordinate of $p$, that is the label of the material point of the reference placement that is placed at $p$ at time $t$. The inertia tensor of the model about $p$ at time $t$ is $I(t,p):=R(t)I_{\nu }(a(t,p))R(t{)}^{\mathsf{T}}\in M_3(\mathbb{R}),$ where $I_{\nu }(a)=\int (\parallel \rho -a{\parallel }_2^2{I}_3-(\rho -a)(\rho -a{)}^{\mathsf{T}})d\nu (\rho )$ is the inertia tensor of the mass distribution about the point $a$ of the reference coordinates (The inertia tensor of a finite bounded Borel measure about any point of ${\mathbb{R}}^3$). Since the label of a spatial point is uniquely determined by the placement, this is well defined; for the centre of mass, $a(t,c(t))=0$ and $I(t,c(t))=I(t)$. The dimension is again $\mu {\mathrm{\ell }}^2$, read in kilogram metres squared.

(d) Moment of inertia about an axis. Let $\mathrm{\ell }$ be a spatial line at time $t$, given by a point $p\in {\mathbb{R}}^3$ and a unit direction $n\in {\mathbb{R}}^3$ ($\parallel n{\parallel }_2=1$). The moment of inertia of the model about $\mathrm{\ell }$ is ${\mathrm{\Theta }}_{\mathrm{\ell }}(t):=n^{\mathsf{T}}I(t,p)n\mathrm{.}$ Its invariance under the choices entering the description of the line and its identification with the integral defining the physical moment of inertia about the line are statements, not conventions; they are proved in clause (c) of the principal-axes theorem of this page. For a body-fixed axis, that is a line through the centre of mass whose direction in reference coordinates is a constant unit vector $n_0$, the moment is $n_0^{\mathsf{T}}I{n}_0=\int \parallel n_0\times \rho {\parallel }_2^{2}d\nu (\rho ),$ by clause (a), and is independent of time, because $I$ and $n_0$ are model data.

(e) What is constant and what moves. The body inertia matrix $I$ is determined by the model's reference mass distribution alone and does not depend on the time, the frame or the motion; the spatial inertia tensor $I(t)$ depends on the motion through the orientation $R(t)$; the inertia tensor $I(t,p)$ about a spatial point depends on the motion through both the orientation and the position of $p$. Principal axes and principal moments of $I$ are the subject of the principal-axes theorem of this page.

• The physical reading of ${\mathrm{\Theta }}_{\mathrm{\ell }}$. The number $n^{\mathsf{T}}I(t,p)n$ is what a consumer reads as the rotational inertia of the body about the line $\mathrm{\ell }$; the identity with the integral of the squared distance from the line is the content of the principal-axes theorem of this page, and is not assumed here.

• Degenerate shapes. If $\nu$ is supported on a line through the origin, then $I$ has a zero eigenvalue, is not invertible, and the relation between angular velocity and angular momentum is not injective; the criterion is proved in Principal axes and principal moments of the inertia data of a rigid-material model, axis moments and extremal values clause (d) below, and the companion page exhibits a two-point-mass body of this kind. The model branch admits such shapes; a consumer that inverts $I$ must state the nondegeneracy hypothesis.

• No measured inertia. The matrix $I$ is model data computed from $\nu$; the item asserts no measured moment of inertia and no claim that any body has a given inertia tensor.