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The reaction moment $\mathbf{m}^{\left(1\right)}$ arising from a torsional damper is $$\begin{equation} \boxed{\mathbf{m}^{\left(1\right)}=\chi_{r}\mathbf{P}\cdot\boldsymbol{\omega}}\,,\label{eq:td-moment} \end{equation}$$ where $$\begin{equation} \boldsymbol{\omega}=\boldsymbol{\omega}^{\left(2\right)}-\boldsymbol{\omega}^{\left(1\right)}\label{eq:td-relative-velocity} \end{equation}$$ is the angular velocity of body $\left(2\right)$ relative to body $\left(1\right)$ , and $\mathbf{P}$ is the projection used in Section 7.9.4.1↑. In particular, $\mathbf{P}_{1}$ is used for joints that cannot undergo relative rotations along any direction, such as prismatic joints; $\mathbf{P}_{2}$ is used for joints that can rotate freely along a single axis, such as revolute, cylindrical and planar joints; in addition, $\mathbf{P}=\mathbf{0}$ for spherical joints. The linearization of $\mathbf{m}^{\left(1\right)}$ is $$\begin{equation} D\mathbf{m}^{\left(1\right)}=\chi_{r}\left(\mathbf{P}\cdot D\boldsymbol{\omega}+D\mathbf{P}\cdot\boldsymbol{\omega}\right)\,,\label{eq:td-linearization} \end{equation}$$ where $$\begin{equation} \mathbf{P}\cdot D\boldsymbol{\omega}=\mathbf{C}^{\left(2\right)}\cdot\Delta\boldsymbol{\theta}^{\left(2\right)}-\mathbf{C}^{\left(1\right)}\cdot\Delta\boldsymbol{\theta}^{\left(1\right)}\label{eq:td-linearization-1st} \end{equation}$$ with $$\begin{equation} \mathbf{C}^{\left(i\right)}=\frac{\gamma}{\beta\Delta t}\mathbf{P}\cdot\mathbf{T}^{T}\left(\boldsymbol{\theta}^{\left(i\right)}\right)\,,\label{eq:td-B} \end{equation}$$ and $$\begin{equation} D\mathbf{P}\cdot\boldsymbol{\omega}=\mathbf{W}\cdot\Delta\boldsymbol{\theta}^{\left(1\right)}\,,\label{eq:td-linearization-2nd} \end{equation}$$ where $$\begin{equation} \mathbf{W}=\begin{cases} \mathbf{W}_{1} & =\mathbf{0}\\ \mathbf{W}_{2} & =\left[\left(\mathbf{n}^{\left(1\right)}\cdot\boldsymbol{\omega}\right)\mathbf{I}+\mathbf{n}^{\left(1\right)}\otimes\boldsymbol{\omega}\right]\cdot\hat{\mathbf{n}}^{\left(1\right)}\\ \mathbf{W}_{3} & =-\left[\left(\mathbf{n}^{\left(1\right)}\cdot\boldsymbol{\omega}\right)\mathbf{I}+\mathbf{n}^{\left(1\right)}\otimes\boldsymbol{\omega}\right]\cdot\hat{\mathbf{n}}^{\left(1\right)} \end{cases}\,.\label{eq:td-W} \end{equation}$$ Combining these results now produces $$\begin{equation} \boxed{\begin{aligned}\mathbf{K}_{mr}^{\left(1\right)} & =\mathbf{0}\\ \mathbf{K}_{m\theta}^{\left(1\right)} & =\chi_{r}\left(-\mathbf{C}^{\left(1\right)}+\mathbf{W}\right)\\ \mathbf{K}_{mr}^{\left(2\right)} & =\mathbf{0}\\ \mathbf{K}_{m\theta}^{\left(2\right)} & =\chi_{r}\mathbf{C}^{\left(2\right)} \end{aligned} }\,.\label{eq:td-stiffness} \end{equation}$$