I had to rush of back to work yesterday and did not complete my reply:
Biomechanics indicate that the assumption of the caster being a source of instability is wrong, so generally speaking, the system is stable (without wind).
Merlin
Please could you explain this comment, I think Gordy is also talking about the same thing. If you are referring to the Milner study the subjects performed targeted wrist movements under conditions where the damping of the wrist was reduced with a load having the property of negative viscosity. In other words the load was only applied in one direction, here would have not been any negative damping.
Both of you are varying zeta to match up your models, which component of zeta are you varying to change the zeta value?
If we use the following definitions:
stability refers to a force that arises depending on the position of the system
damping refers to a force that arises depending on the velocity
Then I would agree that negative damping will only arise as a result of transient negative feedback. Akin to putting a microphone in front of a loudspeaker and taking it away. The only person who would know this would be Gordy. If he said that he always used the same degree of pullback regardless of the acceleration phase, I would accept that as not being negative damping.
However, if it is not negative damping, then you have a transient force that is applied independent of rod tip velocity. However, this force is being applied in anti-phase to the direction of the rod.
Regardless which is correct, if the anti-phase force or negative damping is greater than the opposing damping or opposing rotation force, the rod will go unstable.
stability refers to a force that arises depending on the position of the system
Vince,
It would seem to be that whether or not the rod remains stable as it responds to an external forcing acceleration is applied to the butt of the rod is not very well defined.
What happened to the concept you liked where stability depended on whether a system (in this case the deflection of the rod) returned to some starting position.
If the object eventually does return to its original orientation, then the system is considered dynamically stable.
If there was negative damping in the rod then its deflection amplitude would get larger and larger as the mass velocity increased. That would be an unstable, run-away situation, but show me a case where it actually happens in a fly cast.
Do you have an example where negative damping was induced by some forcing function that had a component that was related to the velocity of the line mass? As shown earlier, just accelerating (or decelerating) the rotation of the butt does not change the sign of the damping coefficient. So what would?
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
If you look at the whole of the defintion that I provided you with, you can see both your quoted statements are true:
Each is concerned with an object's ability to go back to the way it was before a disturbance affected it. Static stability involves only the return of the disturbed object to its original position. Dynamic stability is concerned with how much time it may take for the object to return to its original position.
And if you read my post, I said:
That is also true but the rod does not have negative damping, ccaster does have that abilty because your wrist and arm can go forwards or backwards.
I am struggling to understand why why in every post you appear to state that I am saying that the damping co-efficient of the rod is negative.
Why do you assume that all damping is viscous damping?:
Does your rate of pullback not depend on the velocity of your casting stroke?
The point I am trying to get across is:
A transient damping or external force that is opposing the momentum of the cast, that is sufficiently large will temporarily cause the oscillator to go unstable when zeta is close to zero.
Because the rod is statically and dynamically stable it will return to stability when the force is removed.
If you hit a planet hard enough it will go unstable.
I had assumed that when you used the variations of zeta to match the cast that you were aware that varying the ODE parameters during the cast caused the SHO to act as a parametric oscillator. When Merlin mentioned that he saw a 2F component in the "wave", I had also assumed that we were all talking about phase locking of the input and output due to parametric resonance.
The maths references you seek are here but they were also buried in the Laplace transforms that I provided for you before:
I think this is where my understanding and your diverge:
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter. This effect is different from regular resonance because it exhibits the instability phenomenon.
If you follow the transformation of the equation:
Remarkably, the independent variations g(t) and h(t) in the oscillator damping and resonance frequency, respectively, can be combined into a single pumping function f(t). The converse conclusion is that any form of parametric excitation can be accomplished by varying either the resonance frequency or the damping, or both.
This link provides an explanation of why the phase shift introduces a negative component:
Parametric resonance occurs in a mechanical system when a system is parametrically excited and oscillates at one of its resonant frequencies.Parametric resonance takes place when the external excitation frequency equals to twice the natural frequency of the system. Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter.
The reason that the resonance does not carry on until the rod disappears in a mushroom cloud is:
In parametric resonance the amplitude of the unstable solution grows exponentially to infinity. Damping does not help to saturate this growth contrary to normal resonance caused by an additive driving force. Thus, the nonlinearities in the parametrically driven pendulum are necessary for saturation. This saturation is caused by the fact that nonlinear oscillators have in general an amplitude-dependent eigenfrequency. The growth of the parametrically excited oscillation will shift the eigenfrequency out of resonance.
This mathematical model does draw some interesting parallels:
Parametric excitation differs from forcing since the action appears as a time varying modification on a system parameter.
Vince,
There is no parametric excitation (or pumping) going on in the way we accelerate and decelerate the rod butt. Our forcing functions (especially the deceleration time period) are short, transient, non-symmetrical acceleration functions. That is much different than being some periodic pumping function that has a frequency that is nominally twice the loaded frequency of the rod/line system.
So as that article says our transient non-symmetrical forcing function is much different than superimposing two long-duration periodic excitation functions in order to get the rod to go "unstable."
I had assumed that when you used the variations of zeta to match the cast that you were aware that varying the ODE parameters during the cast caused the SHO to act as a parametric oscillator.
Thanks for doing some research on this, as I was not aware of the instabilities that can come from "pumping" an oscillator. That said I do not think there is any periodic pumping is going on in casting, although it might be interesting so see what happens when you "wiggle" the butt with two precise frequencies.
In my experience when we wiggle the rod butt to excite a second mode oscillation we do not maintain a very precise forcing frequency since the position of the second node moves up and down the rod as our input frequency changes. In addition it would be all but impossible to excite the rod at its natural frequency with one wiggle component and then superimpose another 2x component to pump that oscillation.
But it would be interesting to see what a shaker table could do where the 1x and 2x frequencies could be superimposed as they did in the article you referenced.
We vary zeta in the model just as an approximation of how a firm or relaxed grip might vary the viscous damping in the rod. As said before, I am not sure that is the only thing going on however.
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
In a normal cast I would agree with you and I would not have considered parametric oscillation as a possible standard method of cast analysis. Parametric effects are a problem in helicopter main rotor blades due to control inputs superimposed on the driving frequencies.
However, as a potential root cause of some of the effects we have seen in your pullback cast there are some interesting parallels. In my mind the pullback is a 1 shot pump that effects the damping ratio, a short instabilty followed by self recovery to stability, leading to energetic acceleration at the tip. I can get the 2nd mode excitation on my fibreglass rods quite easily but the 2f "wave" looked like phase locking to the pump to me.
I have spent a while getting my head wrapped around the instabiltiy reqions in Floquets theory and it does seem to be driven by the amplitude of the pump. But as you say the relationships are complex and anything we do here is conjecture
Thanks for drawing our attention on the parametric harmonic oscillator. I am currently reading some papers about that but if we can get something from it, this will be a miracle. It is already quite difficult to explain the SHO, so the parametric version is out of reach for a vast majority of people, even if it is interesting.
Merlin
Fly rods are like women, they wont´play if they're maltreated.
Charles Ritz, A Flyfisher's Life
Not a problem, I understand the difficulty with the application and am not sure that it is worth the effort for the level of fidelity that we could gain. Nevertheless, if I can understand it, I might be able to explain the action by analogy.
However, before we park this, I would like to take you back to the chart from the Floquet Theorem:
What you see is large areas of low frequency instability, I think this relates to long line carries.
You also see instability increasing as you increase the damping ratio at the harmonic frequencies. I think it is important because it is the energy in the unstable regions that power the casts and from where the line speed is derived. You can also see that with a zeta close to 0, you can access the energetic regions with lower levels of forcing amplitude.
If the cast is stable it is tending towards stopping the oscillation so is less energetic.
This is why fighter aircraft a such as the F-22 and Typhoon are designed unstable, this energy is translated to manoeuvrability.
This is true regardless of whether we achieve parametric resonance or not because it is occurs at 1F.
We use the system above "1" in the graphic found in your link, we are faster than the tackle, so we are most of the time on the stable side.
This graphic is just an example. On some scientific papers in the composite domain, the whole thing is more complicate. As usual also , this kind of graphic relates to steady state situations, which is not the case for a fly rod. I never found an example for a transient situation.
For the time being, we can always model close to real casts with a "stable solution". So my conclusion is that we cannot apply this approach today. I mean we even do not have the means to write the correct non linear solutions, and that we do not have any evidence our modeling fails by far. I would not say the same for tip rebound and the like, for which non linearity is likely key.
Merlin
Fly rods are like women, they wont´play if they're maltreated.
Charles Ritz, A Flyfisher's Life
I do agree with what you say and if we had been looking at the paradigm instead of the KK, I am sure we would not have been debating stability as we would not have seen any odd effects. I suspect the stiffer graphite rods are not an issue.
I ran Gordys video past a dynamics engineer and a helicopter flight test engineer, both said unstable. Strangely, the rotor design guys over here are now modelling rotor dynamics based on Floquets Theorm. Unfortunately, I do not keep in touch with this team, I did run some tail rotor trials a few years ago there and wonder if the design came out of this study:
I'm still trying to find out a paper about the transient solution for these equations, without success.
The experts qualified Gordy's (unloaded) cast as "unstable", but did they give a reason why they thought it was unstable? If they were impressed by the speed, there is nothing strange here, the increase in speed can get close to square(2) times the "rigid rod" speed.
Merlin
Fly rods are like women, they wont´play if they're maltreated.
Charles Ritz, A Flyfisher's Life
I'm still trying to find out a paper about the transient solution for these equations, without success.
Merlin,
That is because the transient part of the solution does not reduce to a simple set of real equations.
A numerical solution for a forced oscillator (sinusoidal forcing function) is straightforward to get in MATLAB, but you need to use complex (real and imaginary) values to get it. An example of doing that is shown here. Even after 10 seconds that system had not settled down to the steady state solutions that Vince is talking about.
I expect the chaotic behavior of the vertically driven pendulum is modeled using Lagrangian methods but it is not for the fainthearted that expects to see solutions that involve simple sinusoidal functions. Here is a description of the complexities involved in non-linear dynamics from the pendulum lab web site that Vince referred to earlier.
What is "nonlinear dynamics"? Isn't it a ridiculous term like "non-elephant zoology"? To understand this phrase, imagine the time when programmable computers didn't exist. In those days it was impossible to solve a differential equation like the equation of motion of a pendulum driven by a periodic force. That is, the solution couldn't be expressed in terms of well-known functions like in the case of the linearized equation of motion. Nonlinear equations of motion can be solved only in rare cases. For that reason, physicists tried to build their theories on linear differential equations because they are easier to solve. And indeed, the most successful theories (like electrodynamics and quantum mechanics) are based on linear differential equations. Other even older theories dealing with physical phenomenon closer to everyday experience, like fluid dynamics, were less successful because their dynamics is nonlinear.
Yet, the advent of computers in the last decades made it possible to tackle unsolvable nonlinear problems. This possibility led to a completely different view onto dynamical systems and in association with it to a new language about dynamical systems. The basic terms of this language are more geometrically oriented. Instead of quantitative solutions (which can be obtained only numerically in nearly all cases), qualitative aspects are of greater interest like the type of solutions, the stability of solutions, and the bifurcation of new solutions. Nonlinear dynamics became famous because of the possibility of deterministic chaos, i.e., irregular solutions even though the equation of motion is deterministic. This behavior, that is impossible in linear dynamics, was counter-intuitive and therefore attracts much attention not only by mathematicians and physicists, but also by other scientists and even by the general audience interested in scientific topics. Outside the scientific community, nonlinear dynamics is therefore often called chaos theory, even though not all nonlinear systems behave chaotically.
They have a cool applet for the vertically driven pendulum here that shows the weird behavior you can get with different amplitudes and frequencies for the forcing function.
Anyway I don't think we have to worry about run away fly rods.
Gordy
"Flyfishing: 200 years of tradition unencumbered by progress." Ralph Cutter
It was not the speed, these guys deal with the blade tips in the transonic region so it's not an issue. One of them mentioned that it looked like the onset of blade flutter. I understand what he means and this provides a neat explanation:
In the good old days, everything used to be solved by a few linear equations and proved safe to do lots of trials.
Gordy
Anyway I don't think we have to worry about run away fly rods.
If you've ever tripped over something, stumbled a few paces, but recovered without breaking your nose, or ended up running in a straight line for the rest of your life. You have gone marginally unstable and recovered.
I think that it is likely the amplitude of rod response which makes the system look like an unstable one. Aerodynamics experts see that with their “helicopter blades” eyes, but such amplitude is just the consequence of a transient motion, especially since there is no line on the rod.
I believe we can park this non linear issue about unstable steady state conditions now. The other one is the “instability” mentioned by Paul with the Shakespeare rod, and although experience could tell where the design problem comes from, the theoretical approach remains challenging.
Merlin
Fly rods are like women, they wont´play if they're maltreated.
Charles Ritz, A Flyfisher's Life
