Remember, one cannot measure or even define “rod equivalent mass”. It is just a “fudge factor” which makes the HOE equation “work,” i.e., relates frequency to “added mass.” .
Bill,
If you think the effective mass (mo) is just a fudge factor, you may not be up to date on the subject. We have been able to determine mo from straightforward frequency vs tip mass measurements of a rod for a long time. Now we know mo is related to the KE in the moving mass of a rod when it undergoes a clamped vibration.
Thanks to the insight gained from Dr. Lewin's lecture on how the mass of the spring impacts the freq vs. mass curve of a simple harmonic oscillator, we now have a good understanding of how mo is related to the linear mass density of a rod as discussed here.
This equation is simplicity in itself. F= 0.159*sqrt( K/(M+2.5))
And will not give accurate values for rods that do not have the relatively small mo value of 2.5 grams that you picked for some reason. Also I suspect the K you would use in that application of the SHO equation would be based on your 37% deflection measurements. That will tend to overestimate the frequency value as well.
Here is the measured frequency vs tip mass for a rod whose mo is more than twice that value (6.5 g).
Using the same mo for all rods would lead to predicting much lower loaded frequency values for rods having larger mo values than they would actually have.
Also you will find the spring constant (K) that is used in the frequency vs tip mass equation is not the same as the one you would measure at the 37% relative deflection deflection point you use in CCS. I find the K you get from the frequency vs tip mass equation is closer the the k you get when the perpendicular tip deflection is around 20% of the clamped length as shown below.
Thus I think that the only way to get accurate "effective Ko" and "effective mo" values for the frequency vs tip mass equation is to actually measure how the loaded frequency varies with different tip mass values. Then you would do a least square error fit to that data to calculate the best mo and Ko values
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
To me, there are several parameters which are linked to how one “feels” a rod. The first one is its swing weight (in technical terms, its inertia), because it is a measure of the difficulty there is to move the rod, hence one can feel how difficult or easy it is to rotate the rod, and this increases as you lengthen the line, because the line increases the mass to move. Swing weight is easily detectable by anglers who use to qualify rods as “light in the tip” or at the opposite “tip heavy”. One may appreciate swing weight or not, for heavy and air resisting flies, the “tip heavy” rod is a help. However, when using a 10 feet nymph rod, this can be a worry; you need to privilege the control of the line and fly. Swing weight is also something relative in perception: cast a rod for a #8 line for several hours and then switch for a #4 and then you don’t not care anymore about knowing if this rod has a relatively high or low swing weight per se. Finally, a rod with no swing weight at all has just no interest: no feel means no control, so why making lighter and lighter rods?
The second characteristic that influences the feel is the way a rod bends. You know, some rods just bend down to or under the handle, so up to some point you can get the feel of the bend. You can also visualize the bending effect on loop shape and this can influence your perception of the rod.
The third parameter is the way a rod reacts to a casting motion from the angler, and here you can put the word of frequency. This characteristic is important to understand rod behavior and is high in the ranking place of design for me. I remember my first technical talk with Harry Wilson in 1981, it was the subject of conversation, and all started from there for me. When I went back home Harry asked me to put my ideas on a paper, which he sent to Ed Mosser (competition caster), who forwarded it to Greg Spolek and others. I was tightly in touch with these two people talking about rods and fly casting mechanics for a number of years. By 1987, I produced another one based on the measurements of loaded frequency I carried on my own rods, with the stopwatch of my father and a few lead shots from my fishing bag. The following graphics illustrate these measurements (you can see the poor quality of measurements due to the limited capacity of my measurement tools at that time).
I had curves looking like Grunde’s ones, but I also found the way to convert them in near straight lines, the inspiration came from my books as I tortured the equations. Instead of using the “equivalent mass” concept, I used the slope of these lines and called that the “action parameter”, it is only later on that I changed for “casting parameter”. This slope value is just the reciprocal of the equivalent mass. It is not an elusive concept, but this is just mechanics. It has a meaning and the best example I know is explained during a lesson at MIT (Gordy posted the link) where it is demonstrated by practice without too much theory. This can be computed too, either practically or theoretically and you can get it from any series of frequency measurement for a given rod. The equivalent mass describes the way the frequency of a rod (or any spring) changes with load at tip. So if you can draw a frequency curve for a rod, you can calculate its equivalent mass without any additional experience.
In the equation you suggest, apart the fact that you are using cpm and grams which will affect the unit you have to use for stiffness, you have defined mo as 2.5 grams. So this equation is related to a particular range of rods on the fast and tip action side (that fits western anglers’ taste I believe), and it cannot be considered as “universal”. Why doing so when you have the mean to just calculate mo from your frequency measurements? Here is the trick, and you do not need a special function from Mathlab to find mo: just draw the figure corresponding to the following:
Fo is the unloaded frequency, it is not always easy to capture for fast rods, but as Gordy explained to me, with a fast camera, you can do it with little effort.
Fm is the loaded frequency for mass m, and then you just compute Fo squared divided by Fm squared and you deduct 1 from this ratio. You plot that against m and you get a straight line starting from zero as illustrated in the second picture. The slope is the reciprocal of mo, Excel can do that easily. Just make it with your equation and you will find that mo is 2.5.
But my quest was to understand the physics of casting too and this was the work I was conducting with Ed Mosser. I tried to model the cast and have been struggling at that for a number of years, until the beginning of the 2000 years. I was a subscriber of Fly fisherman Magazine and when Noel Perkins and Bruce Richards published their work about the casting analyzer in December 2003, I just rush and play with my model to mimic their data and found I was correct. I got in touch with Noel for more information. At last my tools where useful so I developed other ones to be able to design rods, using the information given by the casting model. And what does say that model which uses the exact mathematical equations of the harmonic oscillator? It says that there is one single tackle parameter to describe its performance which is its frequency, whatever its stiffness or its own equivalent mass, or load at tip.
I tried to sell the frequency concept to many people, including Harry, but it just failed. Very few rod builders are interested in frequency; they rather rely on stiffness measurements, even if they recognize that this series is more or less faster than that one. To me it is key if I want to design right on the spot. It is not sufficient to properly design a rod, but it is very important, at least if you have a mean to compute the design.
If rod designers are not so interested, what about anglers? When one plays with a casting model, one can see that the maximum line speed is linked to the value of the tackle frequency for a given input (arc, timing, and more precisely, acceleration and deceleration). However, this is not like night and day, the system is tolerant and a small variation in frequency does not lead to a very large variation in speed. At the same time, rod deflection is impacted, and for the “simplest” casting model (linear), the bend is the same whatever the rod stiffness is, it just depends on frequency. The changes on deflection are more visible, the amount of bend is thus an illustration of the effect of frequency (or speed in common language). Now the real rod world is made of non linear springs and that makes some difference at the end of the day (I am currently reviewing that point). We just have drawn a simple picture here with frequency data and a simple harmonic oscillator modeling.
Coming back to feel, I know that one can detect a few cpm difference, but the only “visible” consequence will be on rod deflection. Should we incorporate both elements under the same umbrella wording (speed and deflection)? Should this umbrella be called “frequency”? I understand your question about where can we go with this concept. Not very far I’m afraid. Despite selected publications by Greg and others, there is no response from the rod building industry, and you know why, this is just because design experience can compensate for that up to some degree. The fly rod world is living with that, and maybe this explains the variety of available designs.
I understand the way you are suggesting how to change tackle characteristics bearing in mind the loaded frequency as an indicator. However, as I said, the influence is more visible on bend than on tip speed. I agree with the point that the adaptation is more on the casting arc side than on the load side, even if in practice, the line to rod fit might be a question of line size change for a number of people. Sometimes it is easier to change the line size than one’s casting tuning. I believe that rods bear their designers footprint (or handprint should I say), so if you cast like them, you will appreciate their rods. The problem is that they are fewer designers than anglers.
Let’s face the current situation: CCS has got recognition, it may be desirable to improve it but at the same time any innovation may be disturbing in a world of strong belief. I do not think anglers are able to influence designers to a large extend. That kind of technical approach is more interesting for those who are making their own rods since they try to understand the physics behind, but your experience shows that this is not an easy task.
Looking forward keeping the discussion open
Merlin
Fly rods are like women, they wont´play if they're maltreated.
Charles Ritz, A Flyfisher's Life
Gordy and Merlin,
Thanks for writing your recent comments and for putting up with my games. I believe you are both in agreement that I really don’t know what I am talking about, and you may be correct. However, I must have the last words, so here they are.
For some reason my figure won't copy to this post but here is how you can create it... Plot a harmonic equation. The abscissa - weight is 75 to 180 grains, ordinance (frequency) equals 50 to 150 cycles per minute, plot K every 0.5 units between 0.5 to 1.5) This figure was derived from a plot of the harmonic oscillator formula where m0=2.5 g and K represents the spring factor. What I have done in the following treatment may distress some mechanical engineers. Nevertheless, the conclusions and predictions one can make are close enough to be useful for explaining the interactions of “fly rod feel” and “line weight” to the ardent angler.
To use this chart, I suggest one attach common cents to the rod tip as weights. One cent equals 38.6 grains. Attach 2 cents (77.2 grains) to the tip and determine the frequency. Then add two more cents to make 154.4 grains and determine the new frequency. Plot those two points on the graph and find the curve which best fits those points (interpolate if desired). This curve describes the relationship between frequency and line weight for that particular rod. The angler can then predict what weight of line (or length) will provide his fishing outfit with his own PPF.
