= Ideal rigid link
An \b[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 $\mathbf r_A,\mathbf r_B$,
$$
|\mathbf r_B-\mathbf r_A|^2=\ell^2,\qquad \ell>0.
$$
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.<Advanced Classical Mechanics by Iain W. Stewart>[§1.4, printed pp. 17–19, equations 1.62 and 1.64–1.66]{ref}
Define the axial <unit vector> $\mathbf n=(\mathbf r_B-\mathbf r_A)/\ell$. Differentiating the fixed-length condition gives
$$
\mathbf n\cdot(\mathbf v_B-\mathbf v_A)=0.
$$
Thus 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 $(-0.8,0.6)\,\mathrm{m/s}$ is perpendicular to the link and corresponds to angular speed $2\,\mathrm{rad/s}$. The length stays constant even though the endpoint velocities differ.
A second derivative exposes the required relative <acceleration>:
$$
\mathbf n\cdot(\mathbf a_B-\mathbf a_A)
=-\frac{|\mathbf v_B-\mathbf v_A|^2}{\ell}.
$$
In the example it is $-2\,\mathrm{m/s^2}$ 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.
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