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The virtual work of a connector represents the work of external forces on the rigid body; it is given by $$\begin{equation} \delta G=\sum_{i=1}^{2}\delta\mathbf{v}^{\left(i\right)}\cdot\mathbf{f}^{\left(i\right)}+\delta\boldsymbol{\theta}^{\left(i\right)}\cdot\mathbf{m}^{\left(i\right)}\,,\label{eq:rc-virtual-work} \end{equation}$$ where $\delta\mathbf{v}^{\left(i\right)}$ is the virtual velocity of the joint origin and $\delta\boldsymbol{\theta}^{\left(i\right)}$ is the virtual angular velocity of rigid body $\left(i\right)$ . Using the above relations for $\mathbf{f}^{\left(i\right)}$ and $\mathbf{m}^{\left(i\right)}$ , it reduces to $$\begin{equation} \delta G=\left(\delta\mathbf{v}^{\left(1\right)}-\delta\mathbf{v}^{\left(2\right)}\right)\cdot\mathbf{f}^{\left(1\right)}+\left(\delta\boldsymbol{\theta}^{\left(1\right)}-\delta\boldsymbol{\theta}^{\left(2\right)}\right)\cdot\mathbf{m}^{\left(1\right)}\,.\label{eq:rc-virtual-work-redux} \end{equation}$$ The analysis thus returns the values of the reaction force and moment acting on rigid body $\left(1\right)$ . The virtual velocities at the joint may be evaluated from (7.9-1) as $$\begin{equation} \delta\mathbf{v}^{\left(i\right)}=\delta\mathbf{r}^{\left(i\right)}-\hat{\mathbf{z}}^{\left(i\right)}\cdot\delta\boldsymbol{\theta}^{\left(i\right)}\,,\label{eq:rc-origin-virtual-displacement} \end{equation}$$ where $\hat{\mathbf{z}}$ is the skew-symmetric tensor whose dual vector is $\mathbf{z}$ , such that $\hat{\mathbf{z}}\cdot\mathbf{v}=\mathbf{z}\times\mathbf{v}$ for any vector $\mathbf{v}$ . Substituting this expression into (7.9.1-2) allows us to express the virtual work in terms of the virtual velocities of the centers of mass and the virtual angular velocities of the rigid bodies, $$\begin{equation} \delta G=\left[\begin{array}{cccc} \delta\mathbf{r}^{\left(1\right)} & \delta\boldsymbol{\theta}^{\left(1\right)} & \delta\mathbf{r}^{\left(2\right)} & \delta\boldsymbol{\theta}^{\left(2\right)}\end{array}\right]\left[\begin{array}{c} \mathbf{f}^{\left(1\right)}\\ \hat{\mathbf{z}}^{\left(1\right)}\cdot\mathbf{f}^{\left(1\right)}+\mathbf{m}^{\left(1\right)}\\ -\mathbf{f}^{\left(1\right)}\\ -\hat{\mathbf{z}}^{\left(2\right)}\cdot\mathbf{f}^{\left(1\right)}-\mathbf{m}^{\left(1\right)} \end{array}\right]\,.\label{eq:rc-virtual-work-final} \end{equation}$$