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.

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Projections of sum of all forces acting at section 1-2 to X and Y axes should look like formula 1. Here Nx and Ny are projections of tighting force N to X an Y axes correspondingly. These equations give us the value for cable weight P (formula 2).

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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").

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.

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%).

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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.

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Finally we get (formula 10), where C1 and C2 are coefficients that are defined by point of origin in concerned system. We assume this point to be the lowest point of cable, then C1 = 0 and C2 = 1.

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.

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Cablesag Calculator

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.

We see from Figure 2 that the ratio of tighting force projections (N) is found to be a slope ratio of the force N (see formula 3).

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Using formula 2, we can see that first derivative of projecting of tightening force to Y axis can be showed by the differential of arc (formula 6).

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No Warranties: This calculator and information are provided "as is" without any warranty, condition, or representation of any kind, either express or implied, including but not limited to, any warranty respecting non-infringement, and the implied warranties of conditions of merchantability and fitness for a particular purpose. In no event shall SpaceAge Control, Inc. be liable for any direct, indirect, special, incidental, consequential or other damages howsoever caused whether arising in contract, tort, or otherwise, arising out of or in connection with the use or performance of the information contained on this Web page.

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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)

At the same time, cable weight P is cable weight per unit length (q) mutliplied by differential of arc (dS) (formula 5).

Cable sag (h) is value of cable form equation for point l/2 (formula 12), where l is the straightline distance between the position transducer and the application (Figure 1).

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.

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