Catenary versus parabolic curves: math on the fly
I’ve been tied up this week finishing out the semester as well as assisting with some calculations regarding some equipment to be used in the cleanup efforts in the Gulf of Mexico. In relation to those efforts, I spent some time studying catenary and parabolic curves. Mind you, I was doing this in “crisis mode” at a manufacturing site without many of my reference books and, ultimately, without reliable Internet access.
Specifically I was looking at these from the point-of-view of load calculations and was interested in trying to predict the behavior theoretically (of course, the eventual real-world test didn’t go as planned and my friend’s boat nearly sank because he forgot to put the drain plugs back in after the winter, but, hey…).
So considering some flexible object (like a cable) hanging from two secured points, not necessarily at equal height (e.g. a tow rope connecting a boat to a parasail). Mechanically, the general deflection curve for something like this is given by
where H is the horizontal component of the tensile force in the deflected object and w(x) is a loading function given per unit length. The constants of integration must be determined from boundary conditions.
Suppose we now consider a cable (like a telephone wire) connected at the same height and subjected to a uniform load (e.g. a line of birds sitting on it). The loading function in this case is a constant, the constants of integration can be made to be zero, and the deflection curve turns out to be
which is a parabolic curve. If we assume the greatest sag h occurs in the middle, we set for where is the straight-line distance between the two connection points (i.e. it’s a chord), and thus
We may use this to solve for H and substituting back in we can find the deflection curve entirely in terms of h and l which are both easily measurable:
On the other hand, if we consider a cable loaded only by its own weight, we get a deflection curve
This is actually a catenary curve and not a parabola! To solve for H in this instance, we again set for to get
Unfortunately this is not easy to solve (many mechanics texts simply advise using trial and error on this). Obviously, then, in practice the parabolic curve is much easier to work with. So, given the parameters of the problem under which I was working, I set out to figure out where the two diverged. Using values for w and H from a known model, I overlaid plots of the two curves and found that they began to diverge roughly at a ratio of . This meant that as long as the ratio , I could use a parabola instead of the catenary.
In my real-world model, I knew the value of l. Solving for h from this again required trial and error (at least given that I was doing this in crisis mode on the fly without any of my reference books). So I used a series approximation for a sine function that popped up in the equation and after a few more calculations found that for my real-world application, . Thus I was safe modeling it as a parabola which meant finding other parameters would be much easier!
And that’s “math on the fly,” as it were…