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Equations which describe the sag of the control line.

An excellent way for students to gain a feel for aerodynamic forces is to fly a kite. The motion of the kite through the air is the result of forces being applied to the kite. The kite responds to these forces according to Newton's laws of motion. When the kite is in cruising flight you will notice that the control line produces a gentle curve from the hand of the flyer to the kite as the line sags under its own weight.

There are mathematical equations which describe the sag of the control line, and we can use these equations to determine the altitude of our kite mathematically. A co-ordinate system is established with the X axis along the ground, the Y axis perpendicular to the ground, and the origin in the hand of the kite flyer. The kite is subjected to four main forces; the lift L, the drag D, the weight W, and the tension T in the control line. The control line has a certain length s and the weight of the control line is evenly distributed along the length at p weight per length of line. The total weight of the control line is designated g and is given by:

g = s * p

Because the weight of the control line is evenly distributed along the length, the line hangs under its own weight. There is a differential equation which describes the transmission of forces along a line. The math involved in the derivation and solution of this equation is taught in second year calculus classes and is beyond the scope of this Beginner's Guide. But solving the differential equation, we can relate the shape of the control line (Y versus X) to the distribution of the tension in the control line. The resulting equation is shown in the red box at the top of the page.

Y = C2 + (D/p) * cosh [(p/D) * X + C1]

C1 and C2 are called constants of integration which are determined at the boundaries of the problem. The symbol cosh indicates the hyperbolic cosine function. This is just a tabulated function, like the trigonometric cosine, sine, and tangent, but with different values for a given input. There is also a hyperbolic sine, sinh, and a hyperbolic tangent, tanh, which we won't need for this problem. The equation in the red box is called a Catenary and was developed many years ago to describe cables hanging under their own weight. Other examples would include telephone wires, cables on a suspension bridge, and even a jump rope!

To solve for C1 and C2 we first use the boundary condition that at X=0, Y=0. Plugging X=0 and Y = 0 into the catenary equation gives a relationship between the two constants:

C2 = -(D/p) * cosh (C1)

So if we can determine C1, we now know C2. To get C1 we note that at the origin (X=0) the slope of the control line (dY/dX) is equal to the ratio of the component forces.

At X=0, dY/dX = (L - g - W) / D

Taking the derivative of the catenary equation gives:

dY/dX = sinh[ (p/D) * X + C1]

Plugging in X=0 and the value for dY/dX at X=0 gives the equation:

C1 = sinh^-1[(L - g - W) / D ]

where sinh^-1 indicates the inverse hyperbolic sine of the expression. That just means that the sinh(C1) is equal to (L - g - W) / D. There are tables of inverse hyperbolic sines, just like there are tables of inverse sines in trigonometry. With this value of C1 and the other equation to get C2 as a function of C1, we now know the equation of the entire length of the control line.

The kite flies at some distance XK downwind from the flyer. At the kite we know that the slope of the control line is given by:

At X=XK, dY/dX = (L - W) / D

We can use this information to determine the length XK for a given length of string s. This equation is given at the bottom:

XK = (D / p) * [sinh^-1((L - W) / D) - C1]

The dependence on s is hidden in the constant C1. We could instead write:

XK = (D / p) * [sinh^-1((L - W) / D) - sinh^-1((L - W - p*s)/D)]

If we then plug this value of XK into the catenary equation, we can solve for theheight Y at which the kite is flying above the ground.

Notice that the shape of the line is not a straight line, so we can not use trigonometry to get the height for a given length of line. If the line were straight, we could determine an angle b which is the slope of the line:

dY/dX = tan(b) = (L - W) / D


Y = s * sin(b)

The mathematical equations for the catenary are pretty messy! So we have written a computer program to solve them for you. You can use the KiteModeler computer program to solve these equations for a kite which you can design. You can then build your own kite from your design and compare the actual flight performance to this computer prediction. You can determine the altitude at which you kite actually flies by taking some measurements and using some simple math techniques and a little graph paper. With a little more mathematical knowledge you can calculate the altitude at which the kite is flying. In this case, you can use trigonometry, because it is based on a line of sight which is a straight line.

Enjoy flying ... but always fly safely.


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Editor: Nancy Hall
NASA Official: Nancy Hall
Last Updated: May 13 2021

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