RE: Want to pick your brain on a few things... (PART2)

Discussion of 2nd Mode Response in Fly Casting

I haven’t had time to digest the various e-mails, questions being posed, and outlooks being proposed on how to explain the physical actions//reactions during casting. Hopefully I’ll be able to put concerted effort into that soon. But I thought I would put forward a discussion of the role of 2^nd mode structural response in fly casting. (I am at home for the next two days (June 6, 2009) and have time to devote to fly casting discussions. This first material deals with the 2^nd mode response issue. I also plan to write about mechanisms available to counteract gravity in casting, including aerodynamics. Then having warmed –up, hopefully , I’ll address the casting issues//stop stuff. I should probably proof read this again but am not going to do that so I can finally send it off.)

By the way, as fly casting issues and knowledge surface, I compare them against my current understanding of fly casting physics as expressed in the write-up of this past spring. So far I have seen no reason to modify anything I have written in that document. This write-up hasn’t seen the level of personal review that the physics write-up did so I reserve the right to modify it if further examination warrants. The second mode response is one that is interesting because if second mode response occurs in abundance it couldn’t help but confound the interpretation of some experimental data, complicate interpretation of video, and complicate basic interpretation of cause //effect. For modern composite fly rods I have not seen evidence of excessive 2^nd mode response. The 2^nd mode complicates things because the 2^nd mode response causes rod tip deflections contrary to that intended by casters and they occur differently than 1^st mode deflections because time delays associated with dynamic behavior are much shorter for the 2^nd mode than for the 1^st. Consequently the first mode will be behaving in one manner and the second mode will be behaving in a different manner (with a much shorter or perhaps even negligible time delay – quasistatic response). So, during most of rod unloading the caster imposes net torques on the rod in the same direction as when the rod was loaded. However the equations show that the forcing function (in the equations I have developed for fly casting the torque does not appear as the forcing function – rather the forcing function is the angular acceleration imposed by the caster on the rod handle). The 1^st mode pays tribute to this change by having the torque applied the handle begin to decrease BUT the algebraic sign of the torque does not change immediately. The 2^nd mode is much faster (larger natural frequency) and follows the changing load with a much smaller delay. So while the 1^st mode contributes to the rod unloading smoothly, the 2^nd mode is superimposed and transmits a shaking feeling to the casters hand.

Basically, as you will get some idea of in reading the text, some level of second mode vibration is unavoidable in fly casting. To quantify its significance the best approach would involve computer simulations based on using the process of modal decomposition and superposition. Since this is not readily available the next best approaches might be 1) examining video data of fly casting and 2) scrutinizing your own casting experiences in light of what I will discuss below. I have done these things in limited quantity in the past and so far have concluded that for modern composite fly rods of approximately 9 ft length that second mode response while it must be present is not a significant influence on rod tip trajectory or maximum tip//line speed. If the fly rod universe is expanded to fiberglass rods (like my old Phillipson fly rod), bamboo rods, long spey rods and the like the same conclusions would be inappropriate.

Probably the best way of getting a grasp of these aspects of system dynamic (i.e., vibrational) behavior is by keeping account of the basic facts below that I’ll try to enumerate. These can then be considered with regard to fly casting.

- Under restricted conditions it is reasonable to consider dynamic (anything that changes with time) structural response to be the superposition (that is the algebraic sum) of modal responses

- When these restrictions apply you can think of the modes as being like the alphabet – that is words are composed from the alphabet and structural response is composed from adding together the dynamic responses of the different modes (mode ~ alphabet character). Each mode has its own associated dynamic response//time history and the response of the structure is determined by adding all the modal responses. The modes have the property that they are “complete” meaning that any shape imaginable can be reproduced by combining them in the proper amounts. For some loads, for example a rifle bullet hitting the end of an imaginary (in the sense that it doesn’t break from the impact) fly rod, the motions would be rather unsmooth and time histories of modes must be combined in order to reproduce the chaotic behavior that is caused by the high velocity impact.

- The restrictions for modal superposition to be applicable are rather severe and in many dynamic situations modal superposition cannot be applied. However for fly casting the restrictions are met well enough that modal analysis provides a solid base for understanding.

- Some restrictions are that “boundary conditions” are constant//consistent and that the system characteristics do not change during the response which are I think are satisfied to a degree that allows us to explain behavior using modal theory. Constant//consistent boundary conditions mean that 1) if points on the body are free to move they must be free at all times and if they are restricted from some type of movement this must always be true and 2) there will be one or more points with applied loads and if there is more than one point then the ratio of the various loads must be the same at all times (at any two times, if the load at one point doubles then the loads at all other points must also double, etc). For the sake of discussing fly casting dynamics, it is best to just put aside aerodynamic forces because they do not change the basic character of motion//vibration. Also we can look at motion with fly line by adding the mass of the fly line to the end of the rod or alternately (for simplicity) just consider the motion of a fly rod with no fly line attached. Under these simplifications, adopted to simplify the analysis without changing the substance of conclusions, the system has loads applied at only one location – the rod handle and the boundary conditions do not change during the response//behavior (meaning that loads are always applied through this single location). Modal methodology is applicable.

- Now I want to talk about the modal responses. I have attached pictures of the mode shapes for the first 3 modes of a uniform beam because they are available from a reference in my library. You will see the obvious correspondence between the first mode and what we consider to be nominal prototypical fly rod behavior. The second mode has the odd characteristic that the tip deflection is in the direction opposite to the indicated by the bending at the rod handle. My sensory interpretation of this is that it contributes to feeling shaking sensations at the hand. Hence for rods which are easily excited in the second mode (slower bamboo, fiberglass, etc) the caster senses the rod shaking.

- For fly casting several things influence the degree to which a structural mode is excited when the rod is subjected to a forcing function. These are 1) the modal participation factor (MPF) and 2) the frequency response function of the mode. The effects from these two factors multiply one another to yield the total effect. I am not going to go into it here but for a fly rod undergoing angular acceleration at the handle the loading looks like a body force loading (a magnet exerts a body force loading as does gravity) which is intensified by how far the points are from the rod handle ( and also is proportional to the mass it acts upon). To have a higher MPF the shapes of the mode shape and load intensity must be similar – consequently the second mode and all higher modes have lower MPFs while the first mode has a larger MPF – for fly rod casting the MPFs drop off rapidly and it is probably extremely difficult for a human to excite 3^rd and higher modes to significant levels. This tends to very significantly increase the significance of the first mode. The other fundamental consideration is the frequency content of the load and the “natural” angular frequencies of the modes. The higher the mode, the larger its natural frequency. For a load which rises and falls slowly the frequency content will be mostly at lower frequencies. Conversely for a load which rises rapidly and falls rapidly the frequency content will contain contributions at higher frequencies. A particular mode responds the most for frequency content at values less than 1.5 times its natural frequency – for higher frequency content the response becomes progressively diminished. A mode responds the most vigorously to the frequency content at values near its natural frequency (we call this amplified response – you may be familiar with the terminology “resonant response” – these are the case where the frequency content is the same as the natural frequency and large deflections and vibrations occur). For frequency content well below the natural frequency (say, one-half and less) the response is quasistatic (the same as it would be if the load was changing very slowly or unchanging). So, for example, driving a slow rod with a load which rises and falls very rapidly yields little response in the first mode - a suitable layman’s explanation would be that the load was applied and removed so rapidly that the 1^st mode did not have sufficient time to react substantially to the peak value of load. The second mode on the other hand, even with its low MPF, had a high enough “frequency response function” that you can feel the rod shake in your hand. So summarizing, the rod (and you) would like its first mode to carry the bulk of the response. The matter of timing, speed, and frequency content depends on the relationship between the loading function and the rod’s modal frequencies.

- So now we can get to some conclusions. If casting a rod with a lower first mode frequency ( for example many bamboo rods, the hexagraph rod, etc) it is possible to

1) load and unload the rod quickly enough that the response of the first mode can be significantly reduced. An extreme example – imagine a long greenheart rod and apply a high frequency load by taping the handle end with a steel hammer. You will see very little 1^st mode (or 2^nd for that matter) response because most of the frequency content is so high and can excite only higher modes

2) as casters we seek our rod’s response to be dominated by the first mode. Higher modes produce less in the way of tip/line speeds and are undesirable. If there is considerable higher mode response exhibited it will feel like the rod is vibrating in your hand (in part because what is happening at the rod tip is in the opposite direction as what is happening at your hand.

3) if your are casting a slower rod you still would like the response to consist mainly of 1^st mode response so it is necessary to take more time to move through the casting motions//trajectory which results in creating a frequency content which is lower than that associated with crisper, more rapid motions. Unloading occurs more rapidly than loading so doing things to increase the time it takes to unload might improve matters for the rods with lower natural frequencies. Pushing harder during unloading so it takes longer to unload? However completely eliminating 2^nd mode response may require such low accelerations and reduce the peak loads so much that the response in the first mode is also less than desireable.

4) I can’t see all of the details from the picture of Jason casting but it looks like a) the first mode has unloaded and the rod is in counterflex judging from the rod bend at the handle (perhaps early in the counterflex phase) and b) the 2^nd mode tracks the mathematical loading with relatively ( the physical loading on the rod is the torque applied by the caster however the mathematical loading is a different quantity involving the angular acceleration of the rod butt) small delay time. Thus in terms of what the 2^nd mode has done during the backcast – 1) increased during rod loading yielding deflections in opposite direction as nominal rod deflection, 2) during rod unloading the mathematical forcing function changes direction and the second mode reverses its shape. If the 2^nd mode has a high enough frequency it can track//follow the mathematical loading and could conceivably add to the tip speed of the rod. When the rod goes into counterflex the mathematical load again changes sign and the second mode takes the shape shown in the image. All in all it may not be that important – at best if your’re lucky a little extra tip speed. I’m sure you experimented with the timing in your casts and found the sweetest spot for making the shadow cast but apparently had to put up with the rod shaking at unloading.

5) The significance of the 2^nd mode for modern graphite rods doesn’t seem be overly significant based on the video images available. (That is, I’ve never seen much indication of 2^nd mode presence). If studying bamboo rods it could be different (we had a couple of garage sale rods which in retrospect had lots of 2^nd mode response, it was impossible to eliminate the second mode response and they cast very poorly). The tradeoff between generating large rod loading and minimizing 2^nd mode response in slower rods would be interesting.

6) I can go through a time line of casting and discuss 2^nd mode response that will be occurring during the cast. It is best to describe this in terms of the casting phases I referred to several years ago (I don’t think I referred to these in the last physics write-up that extensively). First of all there is the rod preload phase – here the frequency content is at low frequency and the rod deflection at any particular instant is proportional to the load//moment applied at the handle (i.e., quasistatic). There will be relatively little 2^nd mode response. You will recall we want the rod preloaded so that when higher caster loads are applied (say by rotations using the wrist) the rod tip will not flex backward extensively, but will hopefully move forward at ever increasing speed. Peak rod deflections and peak rod butt angular velocities will occur at close to the same time. At this time rod unloading begins – rod butt angular accelerations change their algebraic sign (meaning that the rod butt angular velocity starts to decrease), the equations also show that forcing functions have changed algebraic sign and are assisting rod unloading (during loading something like the opposite was occurring) so rod unloading occurs relatively rapidly. What this should mean is that the frequency content is largest from peak rod load to RSP. This is where you will see the highest levels of 2^nd mode response and the direction is opposite to that during rod loading. It seems like this response could actually enhance cast performance. The 3^rd phase is counterflex and the mathematical forcing function again changes sign and I think that is what is shown in the image of Jason. We have another video image of Jason – the 200fps data taken way back when at Montana State University and that shows some 2^nd mode response at RSP. In general that data didn’t bring up alarm concerning the significance of 2^nd mode response.

There is much more that could be learned about 2^nd mode response in fly rods. Standard structural engineering analysis tools could shed a lot of light on the situation if someone would perform the computer simulations. However this is probably not forthcoming. When I examine casting video data the area of 2^nd mode response is always of interest to me. Casting video using slower rods has not been coming up in my web surfing – I think it would show lots more 2^nd mode response than what I have been seeing. It is probably a very important topic in understanding the limitation of rods whose natural frequencies are low (Spey, for example). Also important to understanding the causes of rod shake.

A final note is that in my most recent write-up on casting physics the 2^nd mode is largely ignored. I think that approach is valid in general and for modern graphite rods is truthful even in the realm of relatively minute details. But hopefully with time some of this will be quantified in more detail. One thing that could be done is to use the 500 fps video equipment while casting a slower rod and examine the data image by image to see how the 2^nd mode changes during the casting phases.

Discussion of 2nd Mode Response in Fly Casting

Basically as you will get some idea of in reading the text, some level of second mode vibration is unavoidable in fly casting. To quantify its significance the best approach would involve computer simulations based on the process of modal decomposition and superposition. Since this is not readily available the next best approaches might be 1) examining video data of fly casting and 2) scrutinizing your casting experiences in light of what I will discuss below. I have done these things in limited quantity and so far have concluded that for modern composite fly rods of approximately 9 ft length that second mode response while it must be present is not a significant influence on rod tip trajectory or maximum tip//line speed. If the fly rod universe is expanded to fiberglass rods (like my old Phillipson fly rod), bamboo rods, long spey rods and the like the same conclusions would be inappropriate.

- Under restricted conditions it is reasonable to consider dynamic (anything that changes with time) structural response to be the superposition (that is the algebraic sum) of modal responses

- When these restrictions apply you can think of the modes as being like the alphabet – that is words are composed from the alphabet and structural response is composed from adding together the dynamic responses of the modes (mode ~ alphabet character). Each mode has its own associated dynamic response//time history and the response of the structure is determined by adding all the modal responses. The modes have the property that they are “complete” meaning that any shape imaginable can be reproduced by combining them in the proper amounts. For some loads, for example a rifle bullet hitting the end of an imaginary (in the sense that it doesn’t break from the impact) fly rod, the motions would be rather unsmooth and time histories of modes must be combined in order to reproduce the chaotic behavior that is caused by the high velocity impact.

- Some restrictions are that “boundary conditions” are constant//consistent and that the system characteristics do not change during the response which are I think are satisfied to a degree that allows us to explain behavior using modal theory. Constant//consistent boundary conditions mean that 1) if points on the body are free to move they must be free at all times and if they are restricted from some type of movement this must always be true and 2) there will be one or more points with applied loads and if there is more than one point then the ratio of the various loads must the same at all times (at any two times, if the load at one point doubles then the loads at all other points must also double, etc). For the sake of discussing fly casting dynamics, it is best to just put aside aerodynamic forces because they do not change the basic character of motion//vibration. Also we can look at motion with fly line by adding the mass of the fly line to the end of the rod or alternately (for simplicity) just consider the motion of a fly rod with no fly line attached. Under these simplifications, adopted to simplify the analysis without changing the substance of conclusions, the system has loads applied at only one location – the rod handle and the boundary conditions do not change during the response//behavior (meaning that loads are always applied through this single location). Modal methodology is applicable.

- Now I want to talk about the modal responses. I have attached pictures of the mode shapes for the first two modes of a uniform beam because they are available from a reference in my library. You will see the obvious correspondence between the first mode and what we consider to be nominal prototypical fly rod behavior. The second mode has the odd characteristic that the tip deflection is in the direction opposite to the indicated by the bending at the rod handle. By sensory interpretation of this is that it contributes to feeling shaking sensations at the hand. Hence for rods which are easily excited in the second mode (slower bamboo, fiberglass, etc) the caster senses the rod shaking.

- For fly casting several things influence the degree to which a structural mode is excited. These are 1) the modal participation factor (MPF) and 2) the frequency response function of the mode. The effects from these two factors multiply one another to yield the total effect. I am not going to go into it here but for a fly rod undergoing angular acceleration at the handle the loading looks like a body force loading (a magnet exerts a body force loading as does gravity) which is intensified by how far the points are from the rod handle. To have a higher MPF the shapes of the mode shape and load intensity must be similar – consequently the second mode and all higher modes have lower MPFs while the first mode has a larger MPF – for fly rod casting the MPFs drop off rapidly and it is probably extremely difficult for a human to excite 3^rd and higher modes. This tends to very significantly increase the significance of the first mode. The other fundamental consideration is the frequency content of the load and the “natural” angular frequencies of the modes. The higher the mode, the larger its natural frequency. For a load which rises and falls slowly the frequency content will be mostly at lower frequencies. Conversely for a load which rises and falls rapidly the frequency content will contain contributions at higher frequencies. A particular mode responds the most for frequency content at values less than 1.5 times its natural frequency – for higher frequency content the response becomes progressively diminished. A mode responds the most vigorously to the frequency content at values near its natural frequency (we call this amplified response – you may be familiar with the terminology “resonant response” – these are the case where the frequency content is the same as the natural frequency and large deflections and vibrations occur). For frequency content well below the natural frequency (say, one-half and less) the response is quasistatic (the same as it would be if the load was unchanging). So, for example, driving a slow rod with a load which rises and falls very rapidly yields little response in the first mode - a suitable layman’s explanation would be that the load was applied and removed so rapidly that the 1^st mode did not have sufficient time to react substantially to the peak value of load. The second mode on the other hand, even with its low MPF, had a high enough “frequency response function” that you can feel the rod shake in your hand. So summarizing, the rod (and you) would like its first mode to carry the bulk of the response because the MPF for that mode is the largest. The matter of timing, speed, and frequency content depends on the relationship between the loading function and the rod’s modal frequencies.

- So now we can get to the conclusions. If casting a rod with a lower first mode frequency ( for example many bamboo rods, the hexagraph rod, etc) it is possible to

3) if your are casting a slower rod you still would like the response to consist mainly of 1^st mode response so it is necessary to take more time to move through the casting motions//trajectory which results in creating a frequency content which is lower than that associated with crisper, more rapid motions. Unloading occurs more rapidly than loading so doing things to increase the time it takes to unload might improve matters for the rods with lower natural frequencies. Pushing harder during unloading so it takes longer to unload? However completely eliminating 2^nd mode response could reduce the peak loads so much that the response in the first mode is also less than desireable.

4) I can’t see all of the details from the picture of Jason casting but it looks like a) the first mode has unloaded and b) the rod next to your hand is being influenced by the 2^nd mode which means you are reacting as if you have a vibrating rod in your hand (I think maybe you can see the rod strains at the handle which your hand has to cause – the strains at the handle would be zero without your hand there). Because the value of the second natural frequency is lower it could conceivably be in the area of significant frequency content (for a modern graphite rod the natural frequency would be much higher and 2^nd mode response would be quasistatic//unamplified) and thus amplified (meaning there is much more of it). It could be that there is no good solution if you seeking to make a longer cast – taking twice as long to make the cast might eliminate much of the second mode but the maximum applied loads could be so reduced as to also reduce the response of the first mode. I’m sure you experimented with the timing in your casts and found the sweetest spot for making the shadow cast but apparently had to put up with the rod shaking at unloading.

6) I can go through a time line of casting and discuss 2^nd mode response that will be occurring during the cast. It is best to describe this in terms of the casting phases I referred to several years ago (I don’t think I referred to these in the last physics write-up that extensively). First of all there is the rod preload phase – here the frequency content is at low frequency and the rod deflection at any particular instant is proportional to the load//moment applied at the handle (i.e., quasistatic). There will be relatively little 2^nd mode response. You will recall we want the rod preloaded so that when higher caster loads are applied (say by rotations using the wrist) the rod tip will not flex backward extensively, but will hopefully move forward at ever increasing speed. The nature of the loading on the rod tends to reduce the frequency content during this phase of casting (peak load generation//peak tip speed generation phase). Peak rod deflections and peak rod butt angular velocities will occur at close to the same time. At this time rod unloading occurs – rod butt angular accelerations are negative (meaning the caster is reducing the rod butt angular velocity), the equations also show that forcing functions have changed algebraic sign and are assisting rod unloading (during loading something like the opposite was occurring) so rod unloading occurs relatively rapidly. What this should mean is that the frequency content is largest from peak rod load to RSP. This is where you will see the highest levels of 2^nd mode response. This is probably consistent with Jason’s picture with the Hexagraph rod. We have another video image of Jason – the 200fps data taken way back when at Montana State University and that shows some 2^nd mode response at RSP. In general that data didn’t bring up alarm concerning the significance of 2^nd mode response.