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Displacement Cable Sag Error Calculator
Deriving the Catenary Curve Equation
A catenary curve describes the shape the displacement cable takes when subjected to a uniform force such as gravity. This curve is the shape of a perfectly flexible chain suspended by its ends and acted on by gravity. The equation was obtained by Leibniz and Bernoulli in 1691 in response to a challenge by Bernoulli and Jacob.
Displacement Cable Idealized As A Catenary Curve
The equation of a catenary curve can be derived by examining a very small part of a cable and all forces acting on it (see Figure 2)
Figure 2 - Forces Acting on a Part of Cable (Section 1-2)
Here h is the sag the cable gets under the action of gravitational force. To simplify, we will examine two points on the cable: points 1 and 2. Let the distance between point 1 and 2 be so small, that cable segment 1-2 is linear. Let dx and dy be projections of section 1-2 length to X and Y axes respectively.
A tightening force is acting at every point of cable. It is directed at a tangent to cable curve and depends only on the coordinates of cable point. Let the tightening force at point 1 be N and that at point 2 be N+dN, where dN is a small addition due to difference of coordinates.
Let P be the weight of cable section 1-2. Weight is directed downwards, parallel to Y axis. Let α be the angle between the X axis and cable section 1-2.
For cable section 1-2 to be at rest and equilibrium with the rest of cable, forces acting on this section need to balance each other. The sum of these forces need to equal to zero.
Table 1: Derivation of the Catenary Curve Equation
Proving the Calculator
Now some test to prove our calculator above. The input data we have is:
For these default inputs, we can use formulas 7-14 to calculate the cable sag and cable length:
Because the mass of the cable per unit length is so small and the cable tension is relatively high, cable sag does not produce any significant error unless the cable length is exceptionally long (over 60 feet (18.28 meters)). The cable sag error is minor compared to other error sources (generally less than ± 0.0025%).
The easy-to-use calculator above shows how displacement cable sag affects the accuracy of our position transducers. The calculator displays the cable sag in absolute units as well as a percentage of total cable length ("measurement error").
There is virtually no cable sag error when the displacement cable has no appreciable "side loads" on it such as what exists in a space environment or when the cable is oriented parallel to the direction of gravity.
Other catenary facts:
Additional information on the catenary curve can be found at:
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