Ideal rigid link

An ideal rigid link constrains the distance between two attachment points to a fixed length while transmitting an axial constraint force. In the model used here the link is massless, its end joints rotate freely, and it carries tension or compression without bending or stretching. For endpoint positions ${\mathbf{r}}_A,{\mathbf{r}}_B$,This is a holonomic constraint, with reactions that can be represented by Lagrange multipliers. The virtual-work treatment of constrained motion distinguishes these reaction forces from the applied forces that remain in reduced equations.^[ref]

Define the axial unit vector $\mathbf{n}=({\mathbf{r}}_B-{\mathbf{r}}_A)/\ell$. Differentiating the fixed-length condition givesThus endpoint velocities have equal axial components, while transverse relative motion can rotate the link. For an original planar example, let ${\mathbf{r}}_B-{\mathbf{r}}_A=(0.30,0.40)\mathrm{m}$, so $\ell =0.50\mathrm{m}$ and $\mathbf{n}=(0.6,0.8)$. The relative velocity $(-0.8,0.6)m/s$ is perpendicular to the link and corresponds to angular speed $2rad/s$. The length stays constant even though the endpoint velocities differ.

A second derivative exposes the required relative acceleration:In the example it is $-2m/{\mathrm{s}}^2$ along the link. Preserving length does not permit setting the relative acceleration to zero during rotation.

For axial force $+T\mathbf{n}$ on A and $-T\mathbf{n}$ on B, their total actual power is $T\mathbf{n}\cdot ({\mathbf{v}}_A-{\mathbf{v}}_B)=0$. Individual powers can be nonzero, giving constraint power transfer. A freely hinged rigid strut can carry either sign of axial force in this idealization; a taut ideal string can transmit tension but cannot enforce fixed separation by compression when it goes slack.

A real linkage approximates the model only while axial extension, joint friction and link inertia are negligible for the chosen motion. Tracking endpoint separation during loading tests the assumed length constraint; a load cell and position measurements can test the associated force and power balance.