Material Damping in Graphite Rods - What is Rayleigh Damping?

[gordonjudd]

This topic came up in the SHO model trends thread that got sidetracked into a discussion on the damping term used in Merlin’s model. After some research Merlin posted a very interesting (at least to Vince and me) on how the damping ratio used in the model depends on the mass and stiffness parameters of the rod. Rather than having it buried in the SHO model thread I thought it would warrant its own thread.

This is probably of interest to at least a few readers of the technical forum that want to know more about the material science involved with fly rods. However, It has very little practical impact on how the rod behaves from RSP0 to RSP1 since the damping ratio is so small in graphite rods (.02-.03). Consequently the material damping does not have much impact on the spring velocity of the rod as it unloads from MRF to RSP1, but may give some insight on the factors affecting the damping of higher modes in the rod.

If you are interested in some technical mumbo-jumbo, then read on. At present I do not understand what Rayleigh Damping is about, and thank Merlin for giving more details on how it is applied to fly rods. You may have some questions on how it applies as well.

Here is Merlin’s post to start this thread.
Rayleigh coefficient is a very interesting stuff. I took some time to investigate that and here are the findings. First, this is related to internal damping, and the damping ratio can be written as

Damping ratio = alpha /omega /2 + beta*omega /2

While the damping coefficient is = alpha * mass + beta * stiffness.

OK, so I played around with actual values, like for example what we get from the vibration damping of the first mode to try finding out alpha and beta values. The fact is that it is not straightforward and in practice this needs quite complex calculation for structures when you know their harmonics. You have got to use a number of harmonics to get a stable and unique solution for alpha and beta. Here is an illustration extracted from a publication, the “frequency” is in rad/sec, and the range corresponding to fishing rods is between 7 rad/s and 16 rad/s approximately.



Nevertheless we can have a look at the possible ranges for alpha and beta. Let’s look at what I got for a typical 9 feet rod designed for a number 5 line:
If alpha = 0, beta = 0.0032
If alpha = 0.39, beta = 0.0016
If alpha = 0.78, beta =0

We may have rods sensitive to mass damping, others to stiffness damping, I cannot tell today but this opens a new area for investigation. Using the three first harmonics of a typical rod I can say the following:

In the first case, where stiffness damping is dominant, the damping of the third NF can be significant (damping ratio above 0.15), especially when the rod is not loaded (state 2). That makes me think about very fast rods when you just kick them a little bit in the shop they dampen quickly.

When mass damping prevails, this is just the opposite, higher NFs have less damping, very small indeed (0.004). Is this a source for wobbling?

In the intermediate situation, the stiffness character is dominating. So the question is: where are our rods?

If we look at the first harmonic, the values can range from 0.01 to 0.03 for the damping ratio without load. Still small, and when we take mass on board, let’s say 60 feet of line for some 20 grams, the maximum value is 0.78 *0.023 (mo is included) = 0.018 for the damping coefficient, which corresponds to something like 0.07 for the damping ratio: far from being critical (1).

The conclusion for me is that this approach is interesting for state 2 vibrations problems. Thanks a lot for bringing that on the table. For state 1, the drag effect is dominant, as we can see below.

I estimated the effect of air drag on the rod close to RSP, when tip speed is maximum. That can give a damping coefficient about 0.065 for a maximum tip speed of 25 m/s. Now if I imagine the mass being 13 grams and the stiffness 0.74 N/m, that would correspond to about 0.33 for the damping ratio at RSP. Pretty high. At the beginning of the cast the damping effect is nil (no speed), so it is not easy to define what could be an “average” damping ratio for a cast, since the SHO model needs one single value.

The conclusion of this is that when you witness a more pronounced damping effect as you cast a heavier line, it could well come from air drag since you use to cast such line at higher speed. At the very beginning of state 2, we have the same influence of air drag, and it disappears at MCF. After that we enter the material damping effect area again.

Merlin

[gordonjudd]

First, this is related to internal damping
and
The conclusion of this is that when you witness a more pronounced damping effect as you cast a heavier line, it could well come from air drag since you use to cast such line at higher speed. At the very beginning of state 2, we have the same influence of air drag, and it disappears at MCF.

Merlin,
Isn't air drag on the line (or rod) an external force? How does such an external force get entwined with Rayleigh damping if it has to do with "internal" damping?
Gordy

[VGB]

Merlin

In your analysis of aerodynamic loads, you might find this analysis useful as a comparator:

http://www.virtualv8.com/freport.htm

I have not read the detail yet so can offer no comment.

regards

Vince

[Merlin]

Hi Gordy

Maybe I was not clear enough on this point: there must not be confusion between internal damping (Rayleigh coefficients) and aerial damping. I should have written two posts separately. So this post will be made of two parts.

Internal damping

Rayleigh coefficients can be obtained from the set of damping ratio measurements you made with your KK rod; you have two damping ratio values (although there are very close) corresponding to two angular velocities so you can compute alpha (0.1332) and beta (0.00084) from your results.

I applied the conclusion that the damping ratio was not depending on mass at tip to compute what would be the values for alpha and beta for a rod having a constant damping ratio of 0.025 for a couple of measurements; and I get alpha = 0.2547 and beta = 0.00215 for the model rod I chose; which are quite different by comparison to the KK rod, but the basic assumption is slightly crude (damping ratio absolutely constant).

For me it is necessary to investigate what we can find for typical graphite rods, from short (7’) to average (9’) and long to very long rods (DH rods from 14’ to 18’), using a record with no mass at tip top and another record with significant mass, something like 3 times the line mass reference of the rod (e.g. 30 grams = approximately 3 * 9.8 grams for a number 5 line).

What is interesting in the Rayleigh coefficient is the effect of damping on higher modes. The rule of the thumb for such systems (beams) is that the damping value increases with the order of the mode: very high modes are very highly damped. Our interest is rather limited to the second and third mode, so let’s have a look. If you consider 0.025 for NF1 then the damping ratio for NF is the double approximately and you have to double again for NF3, which brings the value around 0.1. In that case, each next NF3 oscillation sees its amplitude divided by 2, and air drag is not taken in account up to that point of the discussion.

External damping

This is air drag effect. I used the last record you sent me (casting a rod without line), and I modeled it in order to find the maximum deflection value recorded. That gave me something like 0.12 for the “average” damping ratioto match your cast. In theory, that is also including a small part coming from internal damping, but if you imagine this is in the range of 0.025 in that case, the part from air drag is 0.095 on average. That’s interesting again. I was not expecting vary large values like what can be estimated at RSP, but something lower which is the case. Incidentally, your cast generates a very high speed (around 48 m/s) for the tip, so a figure of 0.10 should be in the high range. This is four times the value we are using; it corresponds to something like 8 degrees difference in casting arc, not a big revolution in a context of very high speed. The conclusion is that we could move progressively the 0.025 value up as we increase rotation speed (I use about 900 deg/s in this case for the model). A direct link would be easy and in line with equations. If we use 0.025 for let’s say 400 deg/s and 0.1 for 900 deg/s, we just need to interpolate in between.

If you consider the effect of both air drag and internal damping for higher modes, you can understand that these modes can be seriously damped (damping ratio of 0.2 = 0.1 for internal and 0.1 for external damping and above as we approach RSP), another reason not to believe in their contribution to line speed. However, no critical damping before the caster tries to limit counter flex the best he can.

Merlin

[gordonjudd]

Rayleigh coefficients can be obtained from the set of damping ratio measurements you made with your KK rod;

Merlin,

OK, so I played around with actual values, like for example what we get from the vibration damping of the first mode to try finding out alpha and beta values. The fact is that it is not straightforward and in practice this needs quite complex calculation for structures when you know their harmonics. You have got to use a number of harmonics to get a stable and unique solution for alpha and beta.

How do you set up the equations to solve for the alpha and beta terms in that procedure? I have included some better zeta vs tip mass data that you could use as an example in this post.

I went back through that KK data and found the zeta I had computed for the zero tip mass was probably for an older KK rod, not the blank that was used for the 30 gr plot. Here is a re-do of that plot to find the zeta with no tip load for the KK blank. That set of measurements shows the damping ratio does vary with mass, just as the zeta=c/(2.*sqrt(km)) equation says it should.


Here is a table that summarized the data I took when trying to characterize the SHO parameters for that blank.

Fitting the frequency vs tip mass data in that table gives k=2.0 N/m, m0=6.8 g and a natural frequency of 2.72 Hz as shown below.


The second mode showed up in some of the measurements with small tip loads (it did not show up for the 2.5 g load however), and was found to be (3.98 - 4.0) times the fundamental frequency as shown in that table. Based on Haun's measurements (where the third mode was 2.3 times the second mode frequency), I would estimate the unloaded third mode frequency for this blank would be around 4.0*2.3*2.717=25 Hz (157 rad/sec) or 9.2 times the fundamental frequency.

For tip loads above 10 grams the second mode response was buried in the noise, so it would appear the second mode has a fairly large damping ratio for larger tip masses.

Here is a plot of zeta vs the omega for the fundamental mode measured for those tip load masses.


From the table you can see that zeta decreases as the tip mass is increased, so it has a positive slope when plotted against omega.

Here is how zeta varies with the tip mass.


Thanks for taking the time to explain this. I hope the Rayleigh coefficients will explain why the higher modes get damped out of existence with higher tip masses. As you noted in your post that would be yet another reason to doubt that higher modes have much effect on the line velocity.

What are the state 1 and state 2 conditions that you reference in the first post?

Gordy

[Merlin]

Hi Gordy

Here are the equations, using the unloaded rod values as a reference (w0):

beta = ( 2 damp ratio mass m * wm - 2 damp ratio mass 0 * w0) / (wm^2 - w0^2)

and alpha = 2 damp ratio mass 0 * w0 - beta * w0^2

I did that for your data and on average, beta = 0.002335 (range 0.002083 - 0.002485); and alpha = 0.0843

It means that the damping ratio for NF2 = 0.080 and for NF3 = 0.184; so the third NF is significantly dampened. You do not have a perfect straight correlation for w but that may be due to the uncertainties.

It would be nice to have the same kind of results for a glass noodle: beg, borrow, or steal one if you can.

Merlin

[gordonjudd]

You do not have a perfect straight correlation for w but that may be due to the uncertainties.

Merlin,
It appears the values I got for zeta for the 25 g and 30 g tip loads are suspect but it appears the viscous damping value obtained with the Rayleigh approach are quite close to the other tip mass values.

It means that the damping ratio for NF2 = 0.080 and for NF3 = 0.184; so the third NF is significantly dampened.

Do you get those values by multiplying the zeta for the fundamental by the frequency ratio for the other modes? Thus a zeta_1 (for the fundamental mode) = .02 would have a zeta_2 =4*.02=.08 for the second mode, and zeta_3=.02*9=.18?

Is there some other reason the second mode damps out for the higher tip masses? For the 15 g load the zeta for the fundamental mode was smaller than it was for the natural frequency (.017 vs .022). Four times the .017 value would be .068, and yet I do not see any second mode component in that frequency response plot as shown in the region around 6 Hz below.

It would be nice to have the same kind of results for a glass noodle: beg, borrow, or steal one if you can.

Thanks for figuring out how to apply the Rayleigh method. I think the Long Beach Casting Club got rid of the filberglass rods that Jim Green developed at Fenwick for the casting games (cutting edge rods back in the 1960s), but I will see it they still have one in the attic.

Gordy

[Paul Arden]

Hi guys,

apologies for not throwing numbers around :p but I think internal damping may or not be very significant. When I cast 5 weight rods (for example) back-to-back, using the same "hard stop", with some I get tip bounce, with others none. Now I've come to he conclusion that this is hit or miss - for example the Sage XP5 exhibits no significant bounce whereas the Sage ZA5 has (some) bounce - the 4 and 6 doesn't. For me, this is the sole reason the XP5 is a better rod than the ZA5.

My question: is it possible to determine internal damping pre-rod design and somehow therefore manufacture rods to facilitate internal damping or is it just a case of continual trial and error? Currently I believe it's a case of trial and error.

Cheers, Paul

[Paul Arden]

Incidentally I have known for some time that fittings (ie rings) are very important in determining rod response.

Cheers, Paul

[gordonjudd]

It appears the values I got for zeta for the 25 g and 30 g tip loads are suspect

Merlin,
I think the file I had for the 25 gr tip load was for a different mass so I deleted it and got some new values for the c_exp/2 values from the damped cosine fit to the y tip oscillation values.
That gives this new data table:

Those frequency vs mass tip values give
k=1.84 N/m and m0=.0064 kg f_n=2,717

Using your equation for alpha and beta I get
this range of values for alpha and beta for the different omega values:
w-m alpha beta
14.2063 0.2069 0.0019
12.4753 0.0707 0.0024
11.4429 0.1681 0.0020
10.5551 0.0899 0.0023
9.3557 0.1263 0.0022
8.5137 0.0918 0.0023
7.1968 0.0763 0.0024

Alpha appears to be all over the map, but the beta values are reasonably constant. Using the average value of .118 for alpha and beta=.0022 I get this comparison of the measures viscous damping coefficient (blue) with the Raleigh values (green).


Did you see similar jumping around with the alpha values as a function of the omega values you used or did I not use your equation correctly?
I used this code to get the beta and alpha values:
w_sq_dif=omega(2:end).^2-omega(1,1)^2;
beta_all=(2.*zeta(2:m).*omega(2:m)-2.*zeta(1,1)*omega(1,1))./w_sq_dif;
alpha_all=2.*zeta(1,1)*omega(1,1)-beta_all*omega(1,1)^2;
At least the average value resulted in a reasonable match to the measured c values.

What are the state 1 and state 2 conditions that you reference in the first post?

Gordy

[gordonjudd]

My question: is it possible to determine internal damping pre-rod design and somehow therefore manufacture rods to facilitate internal damping or is it just a case of continual trial and error?

Orvis made a big deal about their Trident series of rods that were introduced in 1996 that supposedly used submarine noise cancelling technology to improve the damping. They say:

A vibration dampening system gleaned from the U.S. Navy’s Trident Submarine Program, MVR technology enables the Trident fly fishing rod to cast smoothly by eliminating vibrations, allowing the fly line to move efficiently through the guides. With minimal interference from the fly rod, the the fly line can be directed more accurately.

I just thought those rods were very slow, but I have not seen any measured f_n values that would back that up.

I think the better approach is to lighten up the tip section of the rod as Orvis did with the newer Helios zeries.

Gordy

[Merlin]

Hi Gordy

The basic equation gives
damping ratio = alpha / 2 / w + beta /2 * w

This is the one you solve with 2 determinations of zeta and frequencies.

Beta is more stable so I prefer to get beta first and take an average before calculating alpha.

Merlin

[Merlin]

Paul

Internal damping depends on fibers and resin. The resin itself is something you might change to help damping, but usually you prefer a resin which has high binding properties. If there is some loss in binding somewhere in the structure, then the damping ratio will increase (there is a loss of energy associated).

In theory, you could estimate the damping ratio with a finite element analysis method, but a little bit too much of work for us.

I share your view that all the issue of tip bounce is not only in the damping. It could be linked to an instability of the beam, called "internal resonance". Strange thing indeed. Just an example: imagine you excite the NF2 with your wrist, and that NF3 is a nearly exact multiple of NF2, let's say it equals 2 NF2, then NF3 can pop up at the same time. Controlling that is a nightmare.

Merlin

[Merlin]

Paul

Ring in guides have a role equivalent to mo, this is well knowned. You can compute something like
mo (finished rod) = mo (blank) + mo (guides)

Merlin

[LaMouche]

Ring in guides have a role equivalent to mo, this is well knowned. You can compute something like
mo (finished rod) = mo (blank) + mo (guides)

wraps and varnish are also important (hence the -- justified -- popularity of single foot guides and thin coats)

I certainly understand only a fraction of the discussion, but I wanted to ask for an explaination. I recently build a Batson RX6 blank, and did thorough wiggle tests (had no better means) at various stages of the process. I wanted to know how the system evolves, especially with respect to tip bounce. One of my impressions was that adding mass to the butt (handle and obviously reel) helped a lot with dampening.
now, especially with a bare blank, bounce felt like a wave traveling along the blank, tip down, bouncing at the butt and up again. is that correct? then I guess that ratio between wavelength and rod length should matter a lot wrt tip bounce. is that correct?