To make sense of this thread it would be well to study his excellent analysis of what causes the tip of a rod to whirl when it vibrates. He discusses the physics behind the Lissajous patterns formed by shaft whirl here and goes into to the details of finding the flat line oscillation (FLO) plane here.
Be forewarned if you skip reading his articles the nomenclature and results of this thread will be very confusing for you.
Tutleman reduces the complexity of spine by realizing that to first order it can be reduced to assuming the rod has slightly different spring constants in two primary bending planes that are 90 degrees apart from eachother. The spring constant in the weak or NBP plane is a bit less than it is in the strong or spine plane as shown below.
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For fly rods that stiffness difference is due to the varying amount of scrim overlap along the length of the rod rather than having an oval shape. Graig Spolek has given an extensive analysis of how that overlap effects the spring constant in fly rods in this paper.
But I could not find a free source for it on line.Spolek, G (2005). Measurement of fly rod spines. Proceeding of SEM annual conference on experimental and applied mechanics, Portland, Or, 7-9 June 2005.
In essence the amount of scrim overlap varies along the blank and thus the spine angle orientation varies along the length of the rod as well. Thus its final effect for a full rod is an integrated effect of the varying spine along the length of the blank. Fortunately, to first order at least, the complexities involved with the spine spiraling along the length of the rod can still be reduced to Tutleman’s simple model of two orthogonal springs. It is rather remarkable that something as complex as spine can be reduced to the properties of two orthogonal springs.
In reality the overall spine axes orientation will vary depending on the tip load in the static case or the magnitude and higher mode contributions to the bendform shape in the dynamic case but finding Tutleman’s flat line oscillation planes in a vibrating fly rod can still be used to get a good estimate of the amount of spine in a rod just as it does for a golf shaft.
The key to finding the FLO planes resolves around finding a vibration plane where the angle of the vibration plane is matched to the restoring force given by the combined restoring force of the two spring constants as shown below.
Tutleman shows that the tangent of the angular difference of those angles is given by:
\(tan(B-A) = s *tan(A) / (1 + (1+s) tan^2(A)) \)
Thus the restoring force direction will be in line with the deflection direction when s is zero (no spine), when the deflection angle is at 0 degrees, or when tan(A) has very large values when A is near 90 degrees. Thus even though a rod has a considerable spline it should still have a nearly straight line tip path when its deflection is on the stronger flat oscillation plane or the weaker neutral bending plane.
Taking the inverse tangent of that tan(A-B) function shows how the phase difference between the deflection and restoring angles depends on the relative spring constants in the weak and strong directions as shown in the first plot below:
You can see that for relative spring constant values of around 10% the restoring force will be 2.75 degrees away from the deflection angle when the oscillation plane is half way between the flat line oscillation planes. That phase difference will produce an oval Lissajous pattern that will cause the tip whirl to increase and rotate over time since the oscillating frequency of the two FLO planes is different.
An example of the expected Lissajous pattern expected for a rod with s=.065 and an initial deflection angle of 140 degrees is shown in the second plot below.
Applying this theory to quantifying the magnitude of the spine in a fly rod will be covered in the next post.
Gordy