A spring, damper, and force actuator, acting in parallel between two end frames, is shown in the diagram below.  These three elements are combined into a single translational spring-damper-actuator (TSDA).  

Assuming that the translation vector, $\mathbf{r}$, is directed from frame_a, $\[x_1\,y_1\,$$z_1\]\,$to frame_b, $\[x_2\,y_2\,$$z_2\]$,  the resultant forces are

${\mathbf{f}}_1\=\frac{f}{‖\mathbf{r}‖}\mathbf{\cdot }\left(\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{x1}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{x1}}\+\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{y1}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{y1}}\+\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{z1}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{z1}}\right)$

${\mathbf{f}}_2\=-\frac{f}{‖\mathbf{r}‖}\mathbf{\cdot }\left(\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{x2}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{x2}}\+\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{y2}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{y2}}\+\left(\mathbf{r}\mathbf{\cdot }{\mathbf{e}}_{\mathrm{z2}}\right)\mathbf{\cdot }{\mathbf{e}}_{\mathrm{z2}}\right)$

where ${\mathbf{f}}_1$ and ${\mathbf{f}}_2$ are the reaction forces applied to frame_a and frame_b, respectively and

$f\=\{\begin{array}{cc}{f}_{\mathrm{act}}\+\mathrm{D} \left(\frac{\ⅆ}{\ⅆt}s\right) \+K\left(s-L\right) & s\ne 0\\ 0& s\=0\end{array}$$$

where $s\=‖\mathbf{r}‖$ is the instantaneous length of the TSDA, $L$ is the undeformed spring length, $\mathrm{D}$ is the viscous damping coefficient, and $f_{\mathrm{act}}$ is the prescribed force in the actuator.

Also, ${\mathbf{e}}_{\mathrm{x1}}\,{\mathbf{e}}_{\mathrm{y1}}\,{\mathbf{e}}_{\mathrm{z1}}$ are the unit vectors along the frame_a coordinate frame and ${\mathbf{e}}_{\mathrm{x2}}\,{\mathbf{e}}_{\mathrm{y2}}\,{\mathbf{e}}_{\mathrm{z2}}$ are the unit vectors along the frame_b coordinate frame.

Forces and Moments

Multibody Overview