The conservation of angular momentum is similarly given by $$\begin{equation} \frac{d}{dt}\left(\mathbf{J}\cdot\boldsymbol{\omega}\right)=\dot{\mathbf{h}}=\boldsymbol{\omega}\times\mathbf{h}+\mathbf{J}\cdot\boldsymbol{\alpha}=\mathbf{m}^{ext}\left(t\right)\label{eq:rbm-angular-momentum-balance} \end{equation}$$ where $\mathbf{J}$ is the rigid body mass moment of inertia about its center of mass, $\boldsymbol{\omega}$ is its angular velocity, $\mathbf{h}=\mathbf{J}\cdot\boldsymbol{\omega}$ is its angular momentum, $\boldsymbol{\alpha}=\dot{\boldsymbol{\omega}}$ is the rigid body angular acceleration, and $\mathbf{m}^{ext}\left(t\right)$ is the sum of moments acting on the rigid body. External moments include contributions from user-prescribed moments/torques $\mathbf{m}_{p}^{ext}\left(t\right)$ , from rigid body connectors, $\mathbf{m}_{c}^{ext}\left(t\right)=\mathbf{z}_{c}\left(t\right)\times\mathbf{f}_{c}^{ext}\left(t\right)$ where $\mathbf{z}_{c}\left(t\right)$ is the connector insertion relative to the rigid body center of mass, and rigid-flexible interfaces, $\mathbf{m}_{f}^{ext}\left(t\right)=\mathbf{z}_{f}\left(t\right)\times\mathbf{f}_{f}^{ext}\left(t\right)$ where $\mathbf{z}_{f}\left(t\right)$ is the position of the interface point relative to the rigid body center of mass. Since body forces $\mathbf{f}_{b}^{ext}$ and user-prescribed forces $\mathbf{f}_{p}^{ext}$ act at the center of mass, they do not contribute to $\mathbf{m}^{ext}\left(t\right)$ . Note that $$\begin{equation} \mathbf{J}\left(t\right)=\boldsymbol{\Lambda}\left(t\right)\cdot\mathbf{J}_{r}\cdot\boldsymbol{\Lambda}^{T}\left(t\right)\,,\label{eq:rbm-moi-rotation} \end{equation}$$ where $\mathbf{J}_{r}$ is the mass moment of inertia about the center of mass in the reference configuration and $\boldsymbol{\Lambda}\left(t\right)$ is the rotation tensor representing the orientation of the rigid body at time $t$ , with $\boldsymbol{\Lambda}=\mathbf{I}$ in the reference configuration.

The virtual work statement is given by $$\begin{equation} \delta W=\delta\mathbf{r}\cdot\left(\mathbf{f}^{ext}\left(t\right)-\dot{\mathbf{p}}\right)+\delta\boldsymbol{\theta}\cdot\left(\mathbf{m}^{ext}\left(t\right)-\dot{\mathbf{h}}\right)\,,\label{eq:rbm-virtual-work} \end{equation}$$ where $\delta\mathbf{r}$ is the virtual velocity of the center of mass and $\delta\boldsymbol{\theta}$ is the virtual angular velocity of the rigid body.