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Planar linkages following a prescribed motion

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This reminds me that the (non-existant?) theory of linkage pendulums is still calling my curiosity. Assume the linkage has one angular coordinate $\theta$, which is aligned along the gravitational vertical to give every valid configuration a potential energy value $U(\theta)$. In a first (bad) approximation, we might assume that an overwhelming mass is fixed to the linkage, and that its motion is integrably periodic by simply ignoring moments of inertia for the linkage body. Since the Kempe theorem guarantees a wide variety of $U(\theta)$, we should also get a wide variety of periods $T(\alpha)$, where energy parameter $\alpha$ is fixed by the maximum value of $U(\theta)$. Assuming it exists, what is the algorithm taking $T(\alpha)$ as an input and outputting the linkage associated to the gravitational potential $U(\alpha)$? Can conditions be placed on $U(\alpha)$ to obtain minimal and preferably symmetric linkages?

The main difficulty in theory is that the kinetic term will also depend on the shape of $U(\theta)$, and non-ignorable moments of inertia make this problem even worse. These difficulties are not present in the pendulum case where $\theta$ is truly circular. I've never seen linkage dynamics details worked out, even for a few other special cases such as the Ramanujan periods, which could use more physical intuition toward what they actually are. In practice, it could be difficult to limit frictional loses when the linkage depends on so many bearings. If we bother to calculate models, we would want to follow up with an experiment.

Perhaps the place to start is just choosing a nice simple linkage and get out $T(\alpha)$ or a set of them, then try an experiment to see if energy losses would be prohibitive, maybe not.

In the end it might turn out that some $T(\alpha)$ are not "algebraic enough" (whatever that really means), which could make a good subject for another theorem like Kempe's.

POSTED BY: Brad Klee

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