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By modeling the shape and position of the cable, we are able to optimize the necessary cable length, end point stroke and cable clearance. Note that these same technique can be applied anywhere that the length (or surface area) of an object is known and the amplitude (or enclosed volume) needs to be optimized. Cable assembly design is an obvious application, but others include maximizing the fluid displacement from an infusion pump’s reservoir for a given actuator stroke. In that application, the volume dispensing actuator is often a linear-motion device, while the reservoir is a volume with fixed surface area; by using this simulation technique, the relation between actuator stroke and volume displacement can be uncovered, yielding a smaller, lower-cost actuator to dispense the fluid.

We used MathCAD to model the system:

Because the expression above can be represented as a continuously differentiable vector  function, we can use line integrals to solve for the length of the curve:

The equation for arc length becomes:

From which, we can then plot the shape of the catenary by holding the arc length constant, and determining the catenary parameter for a given cable endpoint.

Plotting the results: y=0 is the vertical position of the moveable endpoint. The “droop” will appear at the y value corresponding to x = 0. The following shows the shape of the half-cable.