Quantify the insensitivity of a rotor to gusts compared to a fixed wing.

[Jean Claude]

A vertical gust of 2.5 m/s, for example, while cruising at 25 m/s, suddenly increases the angle of attack of the rotor disk by Atan (2.5/25) or 0.1 rad.
According my spreadsheet, this momentarily creates an overload of 0.23 g (at constant Rrpm).

Whereas a fixed wing cruising at CL = 0.5, for example, and an aspect ratio of 7, would give a dCL/di slope of 4.5 / rad (*), giving ΔCL = 0.1*4.5 = 0.45, ie an overload of ΔCL / CL = 0.90 g.

The gyrocopter is therefore much less sensitive to gusts than a fixed-wing aircraft.


(*) dCL/di = 5.7/rad + CL/πA If A = Aspect ratio

 

[Rattler 1]

A vertical gust of 2.5 m/s, for example, while cruising at 25 m/s, suddenly increases the angle of attack of the rotor disk by Atan (2.5/25) or 0.1 rad.
According my spreadsheet, this momentarily creates an overload of 0.23 g (at constant Rrpm).

Whereas a fixed wing cruising at CL = 0.5, for example, and an aspect ratio of 7, would give a dCL/di slope of 4.5 / rad (*), giving ΔCL = 0.1*4.5 = 0.45, ie an overload of ΔCL / CL = 0.90 g.

The gyrocopter is therefore much less sensitive to gusts than a fixed-wing aircraft.


(*) dCL/di = 5.7/rad + CL/πA If A = Aspect ratio

I could have told you that without the math.

 

[okikuma]

For some of us, when we visualize a mathematical formula and computation, we also visualize the resultant outcome. That is an image within our "mind's eye" of the mechanical, electrical, or physiological action.

Merci beaucoup Jean Claude.

 

[raghu]

A vertical gust of 2.5 m/s, for example, while cruising at 25 m/s, suddenly increases the angle of attack of the rotor disk by Atan (2.5/25) or 0.1 rad.
According my spreadsheet, this momentarily creates an overload of 0.23 g (at constant Rrpm).

Whereas a fixed wing cruising at CL = 0.5, for example, and an aspect ratio of 7, would give a dCL/di slope of 4.5 / rad (*), giving ΔCL = 0.1*4.5 = 0.45, ie an overload of ΔCL / CL = 0.90 g.

The gyrocopter is therefore much less sensitive to gusts than a fixed-wing aircraft.


(*) dCL/di = 5.7/rad + CL/πA If A = Aspect ratio

JC, I was curious to see how your spreadsheet compares to simple theory. The increase in G load of a rotor due to a gust is given by

deltaG = (GustSpeed * AirDensity * bladeLiftSlope * tipSpeed/bladeLoading)/4

In other words, a gyro behaves like a fixed wing flying at half the tip speed of the rotor and with a wing loading that is the same as the blade loading of the rotor and having the same lift slope as the rotor blade.

Plugging in numbers that are similar to your other examples I get 0.28 g which is close to what you got and I am sure not surprising to you.

 

[Jean Claude]

Raghu,
I suppose your simple theory doesn't take into account the widening of the blades stalled area.
And if we consider torsional effects on the over balanced blades, then deltaG will much less than just from looking at Lock's Number.

 

[raghu]

JC, In general the equation I gave is an upper bound and the simple equation will overestimate somewhat for the following reasons:

1. As you suspected, growing of stalled region is not taken into account in the case of positive load factors, though it probably is not a huge factor in typical cases in gyros.

2. Airfoil pitching moments are assumed to be 0 (as they should be ideally in rotary craft). If the blades are over/under balance, as you say, the load factors due to gusts will be overestimated/underestimated by the simple theory. Does your spreadsheet take blade twisting into account ?

3. A bigger simplification in the theory is that the change in induced flow (due to a gust load) is ignored. This causes the load factor to be overestimated.

It is not hard for the theory to take point 3 into account but it gets a little more complicated to state in a forum post. Doing the calculations with induced flow change taken into account, I get delta g = 0.23 which matches exactly what you got with your spreadsheet (good fortune I suppose ?).

Below, I plotted the case for your example (2.5 m/sec gust, v = 25 m/sec, L =2148N, R = 3.25 m, c = 0.18 m, tipSpeed = 123 m/sec) for various speeds. In the case of the simple theory the g load does not change with speed (orange line) but if we take induced flow into account it does change with speed (blue curve) . Wonder if you get a similar result ?

Finally, in practice gusts encountered are not instantaneous but ramp up to a maximum over some distance thus reducing the overall impact. Not hard to calculate but a topic for later.

 

[Jean Claude]

Raghu, Thank you for your detailed reply.

Does your spreadsheet take blade twisting into account ?

My spreadsheet only takes blade torsion into account if you enter it manually. But this is obviously not possible without knowing their mechanical stiffness.

Doing the calculations with induced flow change taken into account, I get delta g = 0.23 which matches exactly what you got with your spreadsheet (good fortune I suppose ?).

How is it possible to overload without changing the induced speed?. I'm glad we got the same value.

if we take induced flow into account it does change with speed (blue curve) . Wonder if you get a similar result ?

I'll post my curve as soon as I can, for comparison.

 

[Jean Claude]

As promised, here are my results.
6.5 m x 0.18 m rotor, 3.5 degrees of pich setting in flight from level flight lifting 2144 N (395 rrpm at 90 km/h)

The decrease is due to the significant extension of the stalled area on the retreating blade

 

[raghu]

As promised, here are my results.
6.5 m x 0.18 m rotor, 3.5 degrees of pich setting in flight from level flight lifting 2144 N (395 rrpm at 90 km/h)
View attachment 1160996
The decrease is due to the significant extension of the stalled area on the retreating blade

Good stuff, thanks JC! Two comments:

1. At the low speed the induced flow formulas are not very accurate. Also, perhaps we are using different induced flow models. How does your spreadsheet calculate induced flow ? Do you assume constant induced flow or linearly varying from fore to aft of the rotor ?

2. Really interesting to see the huge impact of stalling at relatively low mu. What blade airfoil are you using ? Does you spreadsheet use the cl and cd wind tunnel data for 0 to 90 degrees angle of attack ?

 

[Jean Claude]

At the low speed the induced flow formulas are not very accurate. Also, perhaps we are using different induced flow models. How does your spreadsheet calculate induced flow ? Do you assume constant induced flow or linearly varying from fore to aft of the rotor ?

The average induced velocity is calculated by the simple Prandtls'theory of lifting-line, applied to an elliptical wing (1.27 of aspect ratio)
Hence Vi= Vforward * CN disc / Pi.A .
You can see my results for steady level flight at 30 mph, Average induced speed = 2.16 m/s perpendicular to the disk, at 30 mph. A.o.A = 23.4 degrees, 400 rrpm.


the distribution is assumed to vary from the front to the rear of the disc and depends on the disc A.O.A. (until to obtain constant induced speed when 90° A.O.A.) Affects only b1

What blade airfoil are you using ? Does you spreadsheet use the cl and cd wind tunnel data for 0 to 90 degrees angle of attack ?

Just one general feature: High slope CL/α to max CL, then 2.Sin α Cos α
Slope dCL/dα changing with aspect ratio and Mach number

10 elements per blade + 1 reduced-width element at tip

Extra corrections due to radial flow (Right)

 

[raghu]

Thanks JC for your detailed drawings.

I was curious if a 2.5 m/sec gust does in fact cause stalling in significant portions of the rotor and impact thrust significantly at typical mu values. Modeling rotor stalling is very complex, particularly due to dynamic and transient effects. When AOA rapidly increases airfoils can attain much higher cl than static wind tunnel tests. Also when AOA rapidly decreased from stall it takes a much lower AOA than stall to unstall. Also, non uniform induced flow plays a role.

However, the stall criteria by Bailey and others at NACA years ago is simple to use and has found to give reasonable estimates of when stall becomes a large enough issue to affect performance. Predicting the performance impact is more difficult. This criteria is also useful to know when the theory is valid. Perhaps you are familiar with it ? Essentially, it says for a autorotating rotor, if the AOA is above 12 deg for regions were the blade elements have a tangential velocity > 0.4*tipSpeed then stall will be significant enough factor to affect performance. Otherwise stall can be ignored when calculating performance.

Here is a graph I calculated for the rotor in your example with a 2.5 m/sec gust. As you can tell, in the no gust case, there are large margins from stall becoming an issue in the range mu 0.1 to 0.35. In the case of a sharp 2.5 m/sec gust only at mu =0.35 does stall just begin to be factor.



Below is a more extreme example with a 10 m/sec on the same rotor. In this case stall would be an issue per the criteria across the range of mu above 0.1.

 

[Jean Claude]

When AOA rapidly increases airfoils can attain much higher cl than static wind tunnel tests. Also when AOA rapidly decreased from stall it takes a much lower AOA than stall to unstall. Also, non uniform induced flow plays a role.

I agree. However, the delay in stalling compensates for the delay in unstalling and has no appreciable effect on the rotor's mean CL.

However, the stall criteria by Bailey and others at NACA years ago is simple to use and has found to give reasonable estimates of when stall becomes a large enough issue to affect performance. Predicting the performance impact is more difficult. This criteria is also useful to know when the theory is valid. Perhaps you are familiar with it ?

I'm not familiar with this Naca study you mention, therefore I don't know if they consider Delta g during a gust part of "performances".
Is the Naca report 712 ?.

 

[Jean Claude]

Raghu,
By simply changing the CLMax of the blade sections (and only this parameter) in my spreadsheet, the stalling effect on the delta g produced by the same gust becomes obvious.
Don't forget that the stall aerodynamic angle of the blades is correlated to CL max.
(In this example, almost 20° instead of 12°, which severely changes your sensitivity "margins")

 

[raghu]

Raghu,
By simply changing the CLMax of the blade sections (and only this parameter) in my spreadsheet, the stalling effect on the delta g produced by the same gust becomes obvious.
Don't forget that the stall aerodynamic angle of the blades is correlated to CL max.
(In this example, almost 20° instead of 12°, which severely changes your sensitivity "margins")

View attachment 1161074

Not sure I follow. I am very surprised to see your green points that high, as the maximum cl values are at portions of the rotor that have very low tangential velocity and hence do not contribute much at all to total thrust. Maybe I am missing something ?

The graph I shared is based on classic theory in which, cl max is not considered explicitly(i.e. no stall) while calculating thrust and torque. In general this approach has been shown to give good estimate as long as the AOA < 12 deg at the the radial position where the tangential velocity > 0.4 of tip speed when the blade is at 270 deg (max retreating) position.




As you can see in the graph, 12 deg AOA is only achieved on the 270 degree position at a radial position that has tangential velocity Ut = 0.2 for no gust and mu = 0.25 and Ut = ~0.35 for 2.5 m/sec gust and mu = 0.25 . In other words the theory should give reasonable estimates.

Here is a density contour plot of the blade AOA based on classic theory.


Now, these contour plots assume constant induced velocity and so the max AOA is at the 270 deg position. In practice we know this is not the case but this theory (per numerous NACA papers from the 1940s) in practice been found to give good results.

 

[Jean Claude]

I am very surprised to see your green points that high, as the maximum cl values are at portions of the rotor that have very low tangential velocity and hence do not contribute much at all to total thrust. Maybe I am missing something ?

Most theories implicitly assume that blade sections stall at around 12 degrees. So don't be surprised when my spreadsheet gives such a high delta CL when I enter a stall angle of 20 degrees, deliberately unrealistic. it was just to show the important impact of the stall on the delta g during a gust.

In general this approach has been shown to give good estimate as long as the AOA < 12 deg at the the radial position where the tangential velocity > 0.4 of tip speed when the blade is at 270 deg (max retreating) position.

Yes of course, since stall A.o.A =12 deg is realistic.

 

[raghu]

Most theories implicitly assume that blade sections stall at around 12 degrees. So don't be surprised when my spreadsheet gives such a high delta CL when I enter a stall angle of 20 degrees, deliberately unrealistic

I guess I do not understand what you changed. I am assuming you changed the max cl that the blade element stalls at to 2. Is that correct ?

If so, then that should have very little impact on thrust as only the inner region will be impacted. The theory as such assumes a liner relationship between AOA and cl with no upper bound (stall).

Here are contour plots (math porn ) for only the region where the AOA is between12 and 20.

 

[Jean Claude]

Raghu
Your arguments seem to me to be correct, and I don't understand the discrepancy with my spreadsheet, which nevertheless gives the same results as measurements on PCA2 (Naca report 515).
I was also able to check that the unbalanced torque are in line with the recordings on Cierva C 30 during the take-off run (Aeronautical Research Committy report 1859)
Perhaps my error stems from my assumption that the vertical gust of Z m/s is equivalent to an abrupt change in disc A.o.A of Z / Forward sp. (rad) . But why ?

 

[raghu]

Raghu
Your arguments seem to me to be correct, and I don't understand the discrepancy with my spreadsheet, which nevertheless gives the same results as measurements on PCA2 (Naca report 515).
I was also able to check that the unbalanced torque are in line with the recordings on Cierva C 30 during the take-off run (Aeronautical Research Committy report 1859)
Perhaps my error stems from my assumption that the vertical gust of Z m/s is equivalent to an abrupt change in disc A.o.A of Z / Forward sp. (rad) . But why ?

Ok good! Of course nothing wrong with your assumption of disc AOA increase due to gust, but maybe you can explain in a little more detail how you used that to arrive at instantaneous load/thrust increase. I suspect that will explain the discrepancy. Which graphs did you compare with in NACA 515 ?

The assumptions of the instantaneous increase are:

1. It is a sharp instantaneous gust that changes the inflow of the rotor disc.
2. RRPM and forward speed (and hence mu) do not have time to change
3. But the blades do indeed flap back (a1 increases) to reflect fully the instantaneous conditions

 

[raghu]

Also, JC maybe you are running into issues with number of segments you are breaking the rotor blade into ?

 

[Jean Claude]

Sorry, that was NACA 475
Naca 515 was just to know the actual rotor pitch depending on the load, and to know the chord distribution.
Now I'm going to sleep. Here it is 23:30 o'clock