Body and spatial angular velocity of a rigid-material motion, and the material velocity field

Adopt the model, inertial frame and notation of Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope and Configuration, configuration space and orientation of a rigid-material model in an inertial frame, and let a motion $x(t,\rho )=c(t)+R(t)\rho$ have configuration path $(c,R)$ of class $C^2$ on the time window $I_0$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length).

(a) Body angular velocity. By Rigid placements, body and spatial angular velocity, and material velocities, the matrix $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)$ is skew-symmetric at every $t\in I_0$ (Derivative rules for differentiable matrix-valued curves, and skewness of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ for a curve in $SO(3)$ clause 3), and by The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 2 there is exactly one vector $\omega (t)\in {\mathbb{R}}^3$ with $R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)=[\omega (t){]}_{\times }\mathrm{.}$ The map $\omega :I_0\to {\mathbb{R}}^3$ is the body angular velocity of the motion. Since $R$ is of class $C^2$, the entries of $R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ are differentiable and hence $\omega$ is of class $C^1$; its derivative is the body angular acceleration. The dimension is $[\omega ]={\tau }^{-1}$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework), read in radians per second under the radian convention, in which angles are dimensionless.

(b) Spatial angular velocity. The spatial angular velocity of the motion is ${\omega }_{\mathrm{s}}(t):=R(t)\omega (t)\in {\mathbb{R}}^3,$ of the same dimension ${\tau }^{-1}$. By the conjugation clause of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$, $R^{\mathrm{\prime }}(t)R(t{)}^{\mathsf{T}}=[{\omega }_{\mathrm{s}}(t){]}_{\times }$; this is the space-frame rate matrix of the motion.

(c) Material velocity field. For every material label $\rho$, ${\mathrm{\partial }}_{t}x(t,\rho )=c^{\mathrm{\prime }}(t)+R(t)(\omega (t)\times \rho ),$ because $R^{\mathrm{\prime }}(t)=R(t)[\omega (t){]}_{\times }$ and $[\omega {]}_{\times }\rho =\omega \times \rho$ (Rigid placements, body and spatial angular velocity, and material velocities, The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 1). Writing $p=x(t,\rho )$ for the spatial position of the label and using $R(\omega \times \rho )=(R\omega )\times (R\rho )$, the same identity reads ${\mathrm{\partial }}_{t}x(t,\rho )=c^{\mathrm{\prime }}(t)+{\omega }_{\mathrm{s}}(t)\times (p-c(t)):$ the material velocity field is that of a translation $c^{\mathrm{\prime }}(t)$ composed with a rotation of angular velocity ${\omega }_{\mathrm{s}}(t)$ about the centre of mass $c(t)$. The motion is a pure translation at time $t$ exactly when $\omega (t)=0$, equivalently $R^{\mathrm{\prime }}(t)=0$; then every material velocity equals $c^{\mathrm{\prime }}(t)$.

(d) Uniqueness of the velocity data. The velocity field determines $c^{\mathrm{\prime }}(t)$ and ${\omega }_{\mathrm{s}}(t)$ uniquely: if a motion has material velocity field $p\mapsto v(t,p)=c^{\mathrm{\prime }}(t)+{\omega }_{\mathrm{s}}(t)\times (p-c(t))$, then evaluating at $p=c(t)$ gives $c^{\mathrm{\prime }}(t)=v(t,c(t))$, and for $u\in {\mathbb{R}}^3$ the difference $v(t,c(t)+u)-v(t,c(t))={\omega }_{\mathrm{s}}(t)\times u$ is a linear map of $u$; testing it on two linearly independent vectors determines ${\omega }_{\mathrm{s}}(t)$ by the injectivity clause of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$, and then $\omega (t)=R(t{)}^{\mathsf{T}}{\omega }_{\mathrm{s}}(t)$. The pair $(c^{\mathrm{\prime }}(t),{\omega }_{\mathrm{s}}(t))$ is therefore an intrinsic description of the velocity field; the body angular velocity is its description in the body-fixed frame, and the choice of reference coordinates is part of the model data fixed in Adoption of the rigid-material model branch: reference mass distribution, rigid placements, orientation and scope.

(e) Instantaneous rotation axis. If ${\omega }_{\mathrm{s}}(t)\mathrm{\ne }0$, the line $\{c(t)+s{\omega }_{\mathrm{s}}(t):s\in \mathbb{R}\}$ is the instantaneous rotation axis of the motion at time $t$: it is the set of spatial points whose material velocity equals $c^{\mathrm{\prime }}(t)$, as follows from ${\omega }_{\mathrm{s}}\times ({\omega }_{\mathrm{s}}s)=0$. Conversely, write $w={\omega }_{\mathrm{s}}(t)$ and $z=p-c(t)$. If $w\times z=0$, choose a coordinate $i$ with $w_i\mathrm{\ne }0$; the cross-product coordinate equations (The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ clause 1) give $w_i{z}_j=w_j{z}_i$ for every $j$, hence $z=(z_i\mathrm{/}{w}_i)w$. This proves that the zero relative-velocity set is exactly the displayed line. The velocity field is constant on lines parallel to ${\omega }_{\mathrm{s}}(t)$ and rotates about this axis; if also $c^{\mathrm{\prime }}(t)=0$ at that instant, every point of the axis is instantaneously at rest and the motion is an instantaneous rotation about it. No claim is made that the axis is fixed in time or in space.

(f) Conventions and scope. The convention fixed here is the body-fixed one of the cited mathematical definition: $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}=[\omega {]}_{\times }$ and $R^{\mathrm{\prime }}=R\mathrm{\Omega }$. A consumer using the space-fixed convention must state the change, which is the rotation by $R$ recorded in (b); the two vectors agree only when $R(t)$ fixes $\omega (t)$. Both vectors are kinematic quantities of the motion: no torque, no angular momentum and no equation of motion for $\omega$ is asserted, and no measured angular velocity is recorded. The behaviour of $\omega$ under a change of inertial frame is proved in the frame-covariance theorem of this page.

• The angular velocity is a vector of the spatial space in (b) and of the body in (a). The body vector is the coordinate description of the spatial vector in the body-fixed frame; the identity ${\omega }_{\mathrm{s}}=R\omega$ is the translation between the two descriptions. The distinction matters when a consumer compares the model with data attached to the body, such as principal axes.

• What is used and what is not. Only differentiability of $R$ is needed for existence and uniqueness of $\omega$; the twice differentiability in clause (a) is used for the angular acceleration and, on the dynamics pages of the track, for rate balances. No smoothness beyond $C^2$ across the time window is asserted.

• Degenerate shapes. Clauses (c)-(e) concern the motion of the body as a placement and do not use the mass distribution. The mass distribution enters through the inertia data of Inertia matrix, spatial inertia tensor, inertia tensor about a point and moment of inertia about an axis of a rigid-material model and through the angular-momentum identity of Total momentum, kinetic-energy splitting and centre-of-mass angular momentum of a rigid-material motion, where a mass distribution supported on a line produces a non-invertible inertia matrix.