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The reaction force $\mathbf{f}^{\left(1\right)}$ on a damper is $$\begin{equation} \boxed{\mathbf{f}^{\left(1\right)}=\chi_{c}\mathbf{P}\cdot\dot{\mathbf{g}}}\,,\label{eq:df-damper-force} \end{equation}$$ where $\dot{\mathbf{g}}$ is the time rate of change of the gap function, $$\begin{equation} \boxed{\dot{\mathbf{g}}=\mathbf{v}^{\left(2\right)}-\mathbf{v}^{\left(1\right)}}\,,\label{eq:df-relative-velocity} \end{equation}$$ and $\mathbf{P}$ represents a projection as described in Section 7.9.4.1↑. In particular, $\mathbf{P}_{1}$ is used with spherical and revolute joints; $\mathbf{P}_{2}$ is used with prismatic and cylindrical joints; and $\mathbf{P}_{3}$ is used with planar joints. The parameter $\chi_{c}$ represents the damping coefficient.