Mast Deflection Speeds (esp dinghies)

Mast reflex response.

To elaborate a bit more and particularly comment on the last 2 postings on this thread. Water addict, sorry you are not yet up to speed on this topic. I suggest that you do some more research, and see the link below for a starting point. BWD, you are much more in tune with this under appreciated concept than most of the others. Reflex response is certainly a bigger determinant of ultimate speed through varying wind and water than light weight of spar and stiffness. The proviso is that the boat or board needs to be capable of really significant high speeds, and there is no better example than a well designed sailboard. For explanation of Mast Reflex Response I include this link from Fiberspar, probably the leading windsurfing mast manufacturer in the world. http://www.fibersparsports.com/reflex.html

 

Hi everyone. Tnahks for all your thoughts. It seems that response time is important (at least to high performance boats), and that the faster the better. Calculating the deflection rate by hand seems a little tricky, so I shall look into whether ANSYS can do this.

 

The mast was stored inside during the week between races. The spruce was varnished, can't remember whether the interior of the mast was sealed. It was just the difference in humidity that effected the gust response. The mast was no heavier.

The other difference with carbon Finn masts of the later generation is they are now wing masts with a larger fore and aft cross section, they are much stiffer yet lighter sailors can use them. They are also 8 kg.s versus 12 kg with alloy masts. This combined with plastic sails which are a fraction of the weight of Dacron and the difference is enormous. It has to be rapid response times.
What is interesting is that round skinny masts can be built with the same static bend figures as a wing mast, the usual way to compare Finn/Ok masts. They perform well but whether they have the same response times as the extremely expensive wing mast I very much doubt.

 

Hi Ramona, thats an interesting discussion on the Finn carbon mast. Pretty much what I would have expected. A few years ago very skinny round carbon windsurfing masts came onto the market. As sailboard race sails are very low drag pocket luff leading edge sail shapes, one might expect that a sharper leading edge might lead to higher racing performance. It wasn't borne out in practice, and the very skinny mast has become a rarity now except in waves. Something about it was causing inferior performance to larger sections. Static bend was designed to be similar. The tables for a state of the art standard diameter mast, first table, and the RDM (reduced diameter mast) second table, tell the story. From the Fiberspar web site.
Standard Racing Mast
(cm) IMCS Stiffness Reflex Indicator Carbon Content Weight KG/LBS
430 21 7000+ 100% 1.30/2.86
460 25 7000+ 100% 1.50/3.30
490 29 7000+ 100% 1.67/3.67
520 34 7000+ 100% 1.86/4.09
550 36 7000+ 100% 2.10/4.62

RDM

(cm) IMCS Stiffness Reflex Indicator Carbon Content Weight KG/LBS
370 17 4200+ 95% 1.50/3.30
400 19 4200+ 95% 1.65/3.64
430 21 4200+ 95% 1.90/4.20
460 25 4200+ 95% 2.20/4.80

 

Frosh,
Did some looking on your above mentioned web site. It didn't really provide anything quantifiable other than the natural frequency of vibration - i.e. so what.
Did some further looking on the web, could not find any quantifiable numbers on how rate of deflection of the mast affects average speed on the race course. Help me get up to speed - if you have some documents showing how rate of deflection impacts average speed, I'd like to see them. I could not find any, and am interested in the numbers and methods if you have them.
Thanks.

 

Rate of deflection is the tricky bit. If you were an Olympic sailor then mast manufacturers would probably come knocking on your door, begging you to sail a selection of their products and pick the good ones. Mere mortals have to rely on static bend numbers and take your chances.

Interestingly Mr Elvstrom used to compare wooden Finn masts by placing the bases of the masts together on the ground, then holding both tips just above his shoulder, jiggle both masts the same and sight along them comparing their action. Obviously he understood rate of deflection. I gave it a go and learned nothing from it.

 

Linear beam theory can predict the vibration frequencies of a mast in much the same way that it can predict the static deflection under load. In many ways, a beam vibrates much like a plucked taut string. The dynamic response to a sudden application of a load will result in an initial acceleration, overshooting of the eventual steady deflection, and a damped oscillation as the beam settles down. There will be an infinite number of vibration modes at different harmonic frequencies that sum up to form the dynamic response, although only the first few modes are typically needed to get an acceptable accuracy.

Your question about the time to maximum deflection seems to assume the response will be over-damped and only approach the steady-state value asymptotically from the same direction - this is generally not the case as structural damping tends to be pretty low. There are several parameters you might want to use to describe the response to a sudden step change in the load, including rise time (time to, say, 90% of the steady-state deflection), % overshoot (compaared to the steady-state value), and settling time (when the deflection stays within, say, 5% of the steady-state value). But while these might be easy to calculate for simple beams and measure in a lab, real loadings are not as sharp-edged as this and are more complex. So you might not see overshoot in real life simply because the loading comes on slower than the response time of the mast.

The typical way of calculating the dynamic response is with a finite element method. The elements themselves can be simple beam elements, with the mass of the beam lumped at several nodes along the length connected together by springs that represent the stiffness of each segment of the mast. This forms a matrix equation of the form

M*x_ddot + D*x_dot + C*x = F

where x, x_dot and x_ddot are the vectors of displacements, velocities and accelerations at each node, M is the mass matrix (which also includes mass moments of inertia as well as plain masses), D is the damping matrix, and C is the stiffness matrix. F is the vector of applied loads. The steady-state solution is

C*x_ss = F_ss

So the problem starts out much the same as calculating the static deflection, but now the biggest change is the addition of the mass matrix and acceleration. The stiffness matrix is the same one you use for calculating the steady deflection.

The natural frequency is going to depend on the construction of the mast itself, which will determine how the mast would vibrate if it were floating in space (the free-free modes), and the ways the mast is constrained by the mast step, partners, spreaders, stays, etc. In general, the more the mast is constrained, the higher its frequency is going to be, and the faster its going to reach its deflection due to the load. Of course, the more constrained the mast is, the smaller that deflection is going to be, too.

If you're trying to characterize a mast's dynamic response, probably the best measure is the mast's lowest natural frequency, constrained in some standard way. As you make the mast stiffer and lighter, the natural frequency will go up and it will respond quicker to dynamic loads. For example, you could clamp the mast at the mast step and the hounds, twang the head of the mast and measure the frequency of its vibration. You'll need to do this for both major axes of the mast, since the vibration frequencies will be different in the two axes. If you had an aluminum mast and a carbon-fiber mast of the same stiffness, for example, the carbon mast would be lighter and have a higher natural frequency.

Getting the frequencies and mode shapes right for a complex structure is tricky, even if you have the most sophisticated tools. For example, every new type of aircraft is subjected to a ground vibration test before its first flight. In the GVT, it is suspended by air bags to isolate the landing gear from the ground (so it's constrained as it would be in the air), and shakers are attached to the structure to vibrate it at different frequencies. Accelerometers are liberally applied all over the structure to measure the deflections. The shakers are typically driven either with a frequency sweep (chirp) or with pseudo-random noise that contains a wide band of frequencies continuously. From the structural response to the shakers, it's possible to calculate mode shapes and frequencies, and even back out the mass and stiffness matrices. These are compared with the pre-flight predictions and the models adjusted to match. This is critical for predicting things like flutter speeds and aeroservoelastic instabilities of the control system.

Take a look at
Budiansky, Bernard and Kruszewski, Edwin T, "Transverse Vibrations of Hollow Thin-Walled Cylindrical Beams", NACA Technical Note 2682, April 1952.

Benscoter, Stanley U and Gossard, Myron L, "Matrix methods for calculating cantilever-beam deflections", NACA TN-1827, 1949.

 

Thanks Tom - bonus points for you!
Well, the calcs look a little time consuming for little benefit. So I'll jsut sleep easy knowing that lighter mast respond more quickly, and not try to quantify the effect.

Still, if I get some spare time I might look into tis a bit more.

 

I think you might use natural frequency as an indicator, without actually calculating the transient response. Rather like gyradius is used now for transient motion in waves, and measured with a simple swing test.

However, I suspect that mast natural frequency is going to be correlated with so many other factors, that it's going to be difficult for it to stand out in racing results. Take the aluminum vs carbon mast mentioned earlier. If the two have the same stiffness, the aluminum mast will be heavier (not to mention probably larger in cross section). So if the carbon mast proves faster around the race course, is it because it reacted faster to gusts, or because it provided significantly lower pitching & rolling moments of inertia for the boat as a whole? I suspect the latter is going to be the case.

If you really wanted to find out the effect of mast response, you'd probably have to do a lot of two-boat testing with a carelfully designed dynamic experiment. For example, it may be possible to ballast the carbon mast, say by adding weight at the hounds, so that it didn't affect the boat's gyradius, but still had the difference in natural frequency compared to the aluminum mast. You might have to mount light-weight (styrofoam?) fairings to it to match the cross sections for aerodynamic similarity. Then do a lot of sailing, switching crews to even out the differences there, to get statistically significant results between the two configurations. Naturally, I don't think this is a feasible or even worthwhile program, but it's the only way I can think of to truly answer your original question.

 

Is mathematical proof the only valid methodology?

Hi Water Addict, with due respect to your design qualifications and Tom Speer's Mathematical Analysis, there is much in the high performance sailing arena that relies on empirical semi-scientific method. For instance what was it about Alinghi using calculus or other mathematical model that allowed it to win the latest America's Cup Competition? Sometimes the most effective proof is what actually works on the water reads something like this:

Now, to answer the question "What is the scientific method?" - very simply (and somewhat naively), the scientific method is a program for research which comprises four main steps. In practice these steps follow more of a logical order than a chronological one:

Make observations.
Form a testable, unifying hypothesis to explain these observations.
Deduce predictions from the hypothesis.
Search for confirmations of the predictions;
if the predictions are contradicted by empirical observation, go back to step (2).

 

OK, but I was wondering if you have any documentation showing methods of computation, whether empirical or first principles, relating rate of mast flexure to boat speed?

 

mathematical obsfucation!

Hi Water Addict, in simple terms the answer is that I have never seen a mathematical proof for reflex response being directly proportional to higher boat speeds in very high performance craft. (This is not to say that one does not exist). But why is it necessary? Anyhow my point is that if every empirical observation made in high performance sailing development needed a mathematical model to confirm it, we would all be sitting in university ivory towers, in front of our super-computers, becoming generally very frustrated, with the lack of progress we are making. Instead we can observe what happens on the water, propose a hypothesis as to why is it so, observe if practical events support it most circumstances, and generally enjoy ourselves a hell of a lot more than doing it your way.
I will take the opportunity however of making another point which is very relevant to the discussion; and that is, even if no-one can come out with a mathematical proof as to why Finian Maynard still holds the outright speed sailing record on a sailboard, nothing can not possibly take away the fact that it is so, and that is beyond dispute. BTW, he does not use an alloy mast, nor would he even contemplate doing so, for very good reason, along the lines of my argument. Weight difference is not a factor, I promise you.
I think that you are simply trying to win your argument for no practical purpose, except that you believe that if the other party cannot produce an academically rigorous proof, then their contention has little merit. Sorry, but I have no intention of playing that game with you or anyone. Life is too short for that sort of nonsense.

 

You claimed that I was not up to speed on the topic, so I was looking for your enlightened view to learn more about the topic myself. That's all.

 

Water addict, If you read and absorb the material in my postings and also the stuff about the Finn dinghy masts, and the Fiberspar web site, it is all there for all the interpretation that is necessary to understand the concept. I don't get what the difficulty is with it all? Any way in summary, it goes something like this: A really efficient rig on a high performance dinghy or sailboard needs to be self adjusting and very rapidly so. This means than when the sail is temporarily in a higher than mean wind strength which potentially could produce heeling force that would overcome the human ballasted righting moment, heeling does not occur. You understand that there are gross gusts that can be detected easily by an experienced sailor by seeing cats paws on the water or feeling the higher wind velocity on your face, but also there is an unseen but measurable roughness to what might feel and look like a fairly constant wind. This could be graphed as a low amplitude reasonably high frequency sine wave, graphing actual wind velocity against time. When already fully powered up what then happens when we are on the upper section of the sine wave of wind strength? What needs to happen is that the upper sail twists off very momentarily flattening the upper sail section, and lowering both drag and drive in the upper third, which means that no noticable additional heeling occurs, and the boat accelerates. Possibly a second or two later the sine wave has moved to a point less than the mean wind strength. The mast starts to straighten out in the upper section rapidly, (and this is the entire crux of high speed reflex response), and the sail powers up in the upper third again and the sailor feels almost no change in the heeling force but boat speed barely drops as the rig is suddenly generating more power due to less twist and slightly more camber in the upper third of the sail. This needs to happen very quickly and in harmony with a fairly constant hiking force, i.e righting moment exerted by the sailor, who does not need to rapidly move his body further or less further to windward. I hope that this descriptive but non-mathematical explanation sort of clears up the mystery for you now.