Acceleration curves - different perspectives

Sorry I haven't been reading your emails for the past week. I've been pretty much out of things with the latest flu bug

that's been going around.

The smoothness and acceleration discussion is an interesting one. I'll try to answer the easier question first.

Can an exponential function be smooth? From a mathematical standpoint the answer is yes. In mathematical terms

a smooth function is one for which you can determine the slope (or derivative) of the line at any point on the line. In

layman's terms this means that when you look at a graph of the function it has no sharp corners or breaks in the line.

I've attached a graph of some exponential functions (exponential functions.jpg). As you can see this curve is smooth

from a mathematical standpoint.

From what I've seen of the casting analyzer smoothness is used in a different fashion. It looks at the chart of angular

velocity over time during the positive power application portion of the casting stroke and determines how straight

the line is or how close the curve is to a straight line. I've attached a graph with what we have been calling smooth

(actually constant) acceleration overlaid with the exponential functions (exponential and smooth.jpg).

Could we define an exponential curve that the casting analyzer would consider smooth? From looking at the attached

charts it would be nearly impossible to mistake constant acceleration for exponential acceleration but I note that

the values used by the casting analyzer for expert level smoothness are 2.4 and 5.0 so I believe that it would be

possible to apply exponential acceleration that would fit within the threshhold value of 5.0 for expert level smoothness

but I would need to know the actual calculation used by the casting analyzer to confirm that.

Now the more difficult question - is constant acceleration always the optimal acceleration? In this case we are

calling optimal acceleration to be that which causes the rod tip to follow a straight line path. I believe I can shed some

light on this using mathematics but it is going to take me a few days to put this together.

At this point I would like to offer some anecdotal information. We know that not every rod is the same but an expert

caster can adjust to various rods very quickly and produce expert level casts. This would suggest that, although there

adjustments to be made when changing rods, the changes are well within the limits of very well ingrained muscle

memory (e.g. speeding up or slowing down stroke, adjusting casting arc, etc.). Changing the way in which power

is applied probably doesn't fit inside those parameters. However, I do note that Joan Wulff, who is well noted for

her ability to adjust to various rods states in one of her books that she had great difficulty adjusting to a rod with

a parabolic taper. This would suggest that in at least this one case the rod reacted differently to her casting stroke

than the majority of other rods. We could write this off as a case of a poorly designed rod but on the other hand

perhaps we should consider that any rod chosen should match the caster's natural casting stroke rather than

requiring the caster to spend untold hours practicing with a rod learning how to adjust their stroke to the rod.

That by itself could add a whole new level of complexity to our roles as instructors.

Taking a break for now.

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Walter... Hope you get over your flu bug quickly ! Looking forward to your next message. G.

A couple of non-quantitative thought about acceleration. I do not have any way of measuring the actual acceleration experienced at the rod tip during a cast. I think that our subjective feelings are effected by a lot variables which we are unable to account for. One I have already pointed out. With constant acceleration, during the first 25% of the stroke, time wise, we move the rod tip through 6% of the total stroke length and in the last 25% of the time we move the rod tip through 44% of the casting stroke. I think it is difficult for us to differentiate between acceleration, speed and distance subjectively.

Another is rod angle. During all the discussions of rod angle and how it pertained to fighting fish I was always thinking about how it pertained to the cast. Because we have the rod bent double during the speedup and stop we think we are applying the maximum force. But, we are applying the same amount of force for the same amount of time for the first 1' 5" of a 22 foot casting stoke as we are over the last 9' 8" of that same cast. The rod does not bend as much because it is at a much more effect angle for transmitting force at the beginning of a cast than it is a the end of a cast.

Finally, when thinking about rotational acceleration it is important the consider the effect of rod bending. Rotation predominates during the speedup and stop. The rod transitions from a more efficient force transmission angle to a less efficient force transmission angle during this time period. Therefore, rotational acceleration does not translate directly to rod tip acceleration. Rotational acceleration has to account for increased rod deflection which decreases the acceleration on the line.

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Viewpoints of two highly qualified instructors :

Thanks Mac! I need to catch up on the board and my IM:s, just too much casting going on right now, and you know how that is! (((-:

Before you read this and watch the clips I want to say something. We do all kinds of flycasting and we do competition casting on grass. We still have winter so we cast indoors. Competition casting with light rods looks a bit different from "fishing casting" and not everything we use has a direct application in fishing (stepping around and so on). We do however learn a LOT of things with this that helps us as fishermen, instructors and casters.

I wanted to give Bruce some numbers and we had a indoor session today but no analyser. And let me tell you, measuring that light haired guy's (Fredrik) stroke is downright dangerous with a instrument that has a cable attached to the rod. You have to run around back and fourth and if you get your head in the way of his hand your casting days are probably over. ((((-:

I managed to get some clips initially though. I thought I show you a couple of the competition styles we work on so you all can get a better picture of why we might get steeper acceleration and higher smoothness values. No one had a really good day putting it all togheter today and we are hitting things hangin from the low ceiling all of the time, but this is still a couple of examples. These are casts from +90 ft carries. Turn up the sound so you can hear the power application and the line speed.

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From Thomas Berggren :

Hi Gordy,

Have been away for too long and spent too much time for fishing;-)

However, I?ve tried to catch up this thread and saw my colleague Stefans questions, and all physic links and math . To be honest I can?t follow the math here but it seems rational ??.

Stefans Q:

? When I cast something, a rock, a ball of whatever, I do not feel that the most efficient way to do it is by accelerating the object in a constant manner? I think I peak the acceleration pretty clearly exponential once the cast starts to involve my elbow and wrist joint. Do spear casters accellerate in a constant manner and how about baseball pitchers??

BR?s comment:

? ******The reason that constant acceleration is important in fly casting, and my not be in other types of "throwing", is that we are using a flexible lever in fly casting. To make tight, efficient loops the rod tip must track in as straight a line as possible. How the rod flexes is a major determiner in rod tip path. CA graphs, coupled with high speed video, clearly show that acceleration that is more constant results in rod bend that yields a much straighter tip path than acceleration that is more exponential. The flexible lever is the key, the other sports mentioned don't rely on a flexible lever so how the object is accelerated is much less critical, although I don't have specifics on that. BR ?

My relflection:

That?s right we are using a flexible lever and one of the main thing is to keep a flat tip path. But believe the resistance-change in that lever isn?t always constant, particularly the ?casting machines? with a very stiff butt section. It seems to me that the flex in that lever often peaks (exponential) at the last part of the bending curve.

Just a thought, no math!

So, If I?m consider that it seems rational to me if I have a constant acceleration to the point where the lever resistance starts to peak, perhaps at the same point Stefan mentioned: ? I think I peak the acceleration pretty clearly exponential once the cast starts to involve my elbow and wrist joint ? and from there go: ?pretty clearly exponential ? with my acceleration, still without the result of a ?backlash-bend? that destroy my tip path?

I?m just curious if there is some common sense in this thought?

Regards// Thomas B

Thomas Berggren, Certified Instructor
FFF - Federation Of Fly Fishers,
THCI - Two Handed Certified Instructor.
MCCI - Master Certified Casting Instructor.

Lilla Malma Fishery
Malma

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Thomas ... Makes sense to me as an instructor. I don't know if it will hold up to mathematical scrutiny. G.