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Mathematics and Fly Fishing

Some notes on the mathematical analysis of fly rod action

By D. Y. Barrer

The notes below appear as written by D. Y. Barrer.
There are some errors in Barrer's notes; but he was on the right track.
His substitution results in an easily solved integral for the slope of the deflection curve (3). Unfortunately he make a mistake when calculating the coordinates x and y (equations (5) and (6)).
Barrer's method of calculating the forces acting on the guides is an unnecessary complication.
The moments from the load on the top eye do not change and the contributions from the loads on the guides are very small as the line is nearly parallel to the rod.
The computation (of the loads on the guides) is straightforward although rather tedious, says Barrer. Well, he did not succeed in his program. It is a rather straightforward calculation; but not tedious. We have to keep track of the coordinates of the guides and the angles of the line on either side of the guides as the rod deflects.
When I programmed Barrer's solution I did not include the loads on the guides or the vertical gravitational forces.
The main reason for not including these loads was a wish stay close to Garrison's original work.

Using stored energy, as index does not appear to be useful.

When a load is imposed upon a fly rod during a cast three kinds of forces act ........ tension, torsion and torque. Tension forces act along the long axis of the rod (to lengthen or shorten it), torsion forces act to twist the rod and torque forces cause the rod to bend. Although these forces may react with each other, the tension and torsion forces are usually small in comparison and can be largely ignored. That is, the "action" of the rod is primarily determined by the bend put into the rod during a cast. For our purposes we will take account of both tension and torque forces and ignore any torsion forces.

During a cast some of the work done by the angler is stored in the rod as potential energy some of which is then released to the line as kinetic energy when the rod straightens at the end of the casting stroke. It should be emphasized that all of the energy is supplied by the angler with the rod serving only to modify the manner in which this energy is released to the fly line. To investigate the form of this modification, it is necessary to first compute the shape assumed by the rod when it is subjected to a given force field.

The shape assumed by the rod is governed by the Bernouli-Euler equation:

(1)

Here we have chosen a rectangular coordinate system with the butt of the rod at the origin and the rod extending horizontally in the positive x-direction. E is the modulus of elasticity and I is the moment of inertia in a cross-section about an axis through the center of the rod perpendicular to the long axis. For a rod with a varying cross-section, I varies as a function of x. M is the torque or moment of force at various points along the rod and is also a function of x.

For the calculations it is convenient to take a force field in which there is an acceleration F acting in the positive y-direction and an acceleration G (gravity) acting in the negative x-direction. To determine the moment of force acting at a point Q due to an acceleration F acting at a point P, we proceed as follows: The total moment at point Q is mFx where x is the horizontal distance between the points P and Q. Only the component of the moment normal to the rod at point Q acts to bend the rod. If q is the angle between the tangent to the rod at point Q and the x-axis, this component is mFx cos q . (The tension moment component is mFx sin q ). Similarly, the normal component of the moment due to a force mG acting at point P (in the horizontal direction) is mGy sin q . These two components add to give the total moment normal to the rod at Q due to the two forces acting at point P. See figure (1).

When a fly rod is accelerated during the casting motion there are many different forces acting at many different points along the rod. The acceleration of the line mass extending beyond the tip creates a force at the tip. The mass of the rod itself as it is accelerated gives rise to additional forces as does the line in the guides, the guides themselves, the ferrule, the varnish and windings, etc. To compute the moments for the line in the guides requires knowing the guide spacing for a force on the rod can only occur at the guides. The computation is straight forward although rather tedious. Most of the calculational effort required to solve Equation (1) is involved in computing the moments.

In structural engineering one is usually concerned with beams that flex very little under load. When this is the case, Equation (1) can be simplified by setting dy/dx = 0. Also, under this simplifying assumption the moments are simply computed as mF times the distance along the beam between the point of application P and the point Q. The force mG acts along the beam to generate bending moments equal to zero. However, for a fly rod such simplifying assumptions are not valid and the "cookbook" application of the simplified formula from some engineering handbook can lead to serious and misleading results. Nevertheless, such attempts at analysis have been made in the past; fortunately, they have not attracted much attention and good rod tapers have evolved largely by intuition, trial and error.

Formidable as Equation (1) might appear (a second order, non-linear differential equation), its solution is easily within reach now that computers have become generally available and comparatively inexpensive. One can now purchase a computer adequate for the job for about the cost of one good bamboo fly rod! Software for making the necessary calculations is not available commercially so one must write his own. One purpose of this exposition is to outline the basic mathematics so that one can program the calculations.

A particularly simple, and somewhat idealized, casting model is used for most of the calculations. In this model the angler holds the butt of the rod in a vertical position and moves his hand forward (and backward) parallel to the ground (water) with an acceleration of N times that of gravity. The forces mF (see Figure (1)) are then the various weights of the rod sections and its various components (line, ferrules, etc.) times N. The forces mG are simply the weights of the various parts.

In Equation (1) set dy/dx= tan q to obtain,

or,

(2)

This can be further simplified to

(3)

or,

(4)

For our purposes, the numerical integration will use Equation (2). (If one wanted to effect the solution without a computer, one might select one of the simpler appearing equations.) With the coordinate system we have chosen we know that q lies between zero and 90 degrees and, accordingly, we may be assured than when we want the value of the arctan, the principle value returned by the computer will be correct. (Actually, I tried all three equations to find that with initial values of q all set to zero it always converged. In some cases the other equations did not converge; even when they did, convergence was faster with Equation (2).)

The computer program performs the numerical integration using intervals corresponding to five inch rod segments. Suppose that we have some approximate values of q at intervals of 5 inches along the rod. If we use these values we can hope that Equation (2) will give new values of q that will be more exact. We can then use these new values and repeat the process to obtain even more exact values. There is, of course, no guarantee that this iterative process will converge. Without going into the details of a formal convergence proof, let it suffice to note that experience with a large number of fly rod tapers has shown that values of q correct to at least three significant digits results after five or six iterations when the initial values are all set to zero. Inasmuch as the input values for the weights and rod dimensions are usually known to three significant digits at best, this degree of accuracy is sufficient.

Except for the first iteration, we will need to know the values of x and y. To compute them, let q₁ and q₂ be the angles that the rod makes with the x-axis at the ends of any given 5-inch segment. If we construct a circle tangent to the curve of the rod at these two points, a bit of trigonmetric juggling leads to the expressions:

(5)

should be

and,

(6)

should be

The expression for x gives us a numerical value of dx for the numerical integration and by summing the expressions for x and y we have the proper distances for computing the moments. Rather than use one of the standard algorithms for the integration, we take advantage of the fact than the rod shape is convex and use this circle approximation to compute the area under the curve, thus allowing use of the relatively large 5-inch intervals to obtain sufficient accuracy without inordinately increasing the running time.

If the reader is attempting to program this procedure, it will be helpful to refer to the book "A Masters Guide to Building a Bamboo Fly Rod", by Garrison & Carmichael (Martha's Glen Publishing Co., 1977). Here, correct formulas for the volume and center of mass of hexagonal prisms are given. (The procedure shown for computing stress curves, or that for calculating the taper for given stresses, is not applicable since it uses the "simplified" Bernouli-Euler equation.) To compute the weights of the bamboo segments one could use a variable density curve such as the one given in this book. It has been the author's experience, however, that this makes very little difference in the final results compared with using a constant average density for rods of 8 1/2 feet or less.

The moment of Inertia I, is simply a geometric property of the cross-sectional shape. For a rod with a hexagonal cross-section, it is given by,

(7)

and for a circular cross-section by,

(8)

in which d, the diameter, is the distance between flats for the hexagonal rod. If the rod is hollow, the formulas must be modified by subtracting a similar expression using the inside diameter.

The modulus of elasticity, E, for bamboo is of the order of 108 if we are using units of weight in ounces and lengths in inches. It will vary somewhat between different pieces of bamboo and is affected by the amount of heat treatment as well as the varnish, wrappings, glue, etc. There is a simple way of determining the modulus for a finished rod. If we clamp the rod near the butt in a horizontal position, hang a known weight on the tip and measure the tip deflection, we have enough information to calculate the modulus. Using the "static" model in the program (G=0 and N=1) one can compute the tip deflection for various values of E. A value of E that yields the measured deflection, is a good working value that can be used in the casting (dynamic) model since E is a property only of the material.

The bending stress in the bamboo fibers at a distance c from the central or neutral plane is given by,

Thus, the maximum stress occurs in the outer fibers where c=d/2. For a solid rod with a hexagonal cross-section the maximum bending stress is,

A similar expression is obtaineds for rods of circular cross-section. The total stress can be obtained by adding (algebraically) the tension stresses.

Bamboo used in rod construction can withstand stresses of over 220,000 oz/sq.in. without fracturing. However, before fracture occurs the rod will take a "set". That is, the elastic limits have been exceeded and the rod will not return to its original shape after the loads are removed. The stress at which this occurs is variable. For example, it is quite temperature sensitive. Indeed, the procedure to remove a set from a rod is to apply heat and stress the rod in the opposite direction and in doing this, one must be careful to not apply too much heat. Furthermore, a given stress applied and held for a length of time may produce a set whereas the same stress applied only momentarily may not. Thus, in practice it is prudent to use a rather generous engineering safety factor. The calculation of maximum stresses under casting loads for a number of very servicable rods shows that a factor of at least two is the norm. That is, calculated stresses seldom exceed 110,000 oz/sq.in.

Rod "action" is something we all talk about but is difficult, if not impossible, to define quantitatively. Clearly, there is no "ideal" action. A rod that is the pride and joy of one angler may be an abomination to another. Casting styles differ. Rods that are good for short distance casting will not perform optimally for longer casts. Rods good for handling small flies may not work well with heavy bass bugs. A dedicated fly fisherman will normally have a number of rods in his arsenal and select the right rod for the kind of fishing the situation demands. The number of rods he has will depend upon the variety of kinds of fishing he intends to pursue and the range of conditions that any one rod will handle satisfactorily. If the angler can describe the kinds of uses to which he intends to put a rod, the custom rod maker, if also an experienced fisherman, can select a taper based on experience that will meet the angler's requirements.

If the angler can describe the action he wants by reference to a similar action in another rod, but would like a rod for a different line weight or of a different length, the rod maker has a scaling problem. If, as seems reasonable, the action characteristics are imbodied primarily in the stress curve, the mathematical computations described here can be useful. The evaluation procedure described earlier can be inverted. That is, one starts with a given stress curve (similar to that of the exemplar rod) and computes the taper to yield that curve using the revised rod parameters. The computation time using the computer program is considerably longer. Each diameter iteration will require four or five moment iterations and, to get three significant figure accuracy, as many as ten or more diameter iterations may be required.

As noted earlier, when a fly rod is flexed during a cast, some of the energy supplied by the angler is stored in the bent rod as potential energy. As the rod straightens at the end of the forward stroke and the line is released this energy is imparted to the line as kinetic energy added to the kinetic energy the line may already possess. This total energy determines the initial line speed. There is, of course, some energy loss due to friction and air resistance but, most of the energy is kinetic energy or energy of motion. A comparatively stiff rod used to cast short distances will not flex very much and the energy stored in the rod will be less. A soft rod under the same conditions will flex more and more of the energy will be stored in the rod for later release. If the amount of stored energy is large we describe the action as "slow" and. if small, the action is "fast". Thus, a given rod will have a slower action when casting long distances and a faster action at short distances. It is useful to calculate the amount of stored energy as an index of rod action; the computer program does this. If the line is released at the end of the forward casting stroke at the moment that the rod has completely straightened, the total stored energy is divided between the line and the rod in proportion to their respective masses.

After the line is released, the rod continues to flex in the opposite direction. For this idealized casting model the rod tip will drop below the line. Unless compensated for by the angler, this can produce an undesirable wave in the line. The skillful angler can raise the entire rod to minimize this "wave". It is of passing interest that for most of us this wave should be less pronounced when using fast action light weight rods since less energy remains in the rod and the reverse tip deflection will be smaller. The lighter weight of graphite rods will, also, tend to minimize the line wave.

The program Unirod_Barrer is based on Barrer's notes.
The difference between the UniRod program based on the geometrical properties of the deflection curve and the program based on Barrer's notes is negligible.

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