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Post by Br. Marius on Apr 16, 2015 12:42:33 GMT
Show flap response of a hovering articulated rotor (hinge at rotation axis and no hinge spring) due to sudden exposure to a uniform vertical gust field (Vg constant). Assume only a single degree of freedom flap motion, neglect lag and torsion motions, and include a dynamic inflow model.
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Post by matthorr on Apr 17, 2015 0:42:27 GMT
I don't know if this is in the ballpark or not... Scan 2015-4-16 0008.pdf (307.71 KB). If it is, it seems too easy, so what am I missing? I assumed that \(\Delta C_T = 0\) for the dynamic inflow and just added \(V_g\) to \(\delta U_P\). |
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Post by Br. Marius on Apr 17, 2015 19:07:11 GMT
I'm starting off the same way that you are...but I'd think that we would have to include non-circulatory forces as well. Both kinds are perturbed by a vertical gust, so I'd think that both would need to be included in the derivation.
As for dynamic inflow...I don't know how I'm going to attack that yet. I'd think that \(\Delta C_T\) would be necessarily nonzero, but I don't necessarily know how to relate it back to known values. Actually, the value just relates to your statement about \(u_p\) where the gust velocity just adds to the perturbed value. Hence, the change in CT is just the difference between the current thrust coefficient (expressible in terms of the induced inflow ratio) and the new one, which are related by Vg because the gust would be trying to push a a mass flow back up into the control volume with a force of \(\dot mV_g=0.5\rho A_{rotor}V_g^2\) and would effectively decrease the flow speed exiting the control volume in momentum theory analysis of a hover rotor. Hence, the change in CT becomes something like \(\Delta C_T=\frac{0.5\rho\pi R^2V_g^2}{\rho\pi R^4\Omega^2}\)...maybe?
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Post by matthorr on Apr 17, 2015 20:36:01 GMT
From (trying to) asking Dr Chopra about the last homework, he indicated that the non-circulatory forces are only important/significant for the moment (needed for torsion). I still don't follow his notes how he derives \(L_{\mathrm{NC}}\) in terms of \(U_P\), etc... I don't think you are changing \(C_T\) - that term in dynamic inflow is only used (I believe) to give a lag to the response of \(\lambda\) to changes in T, but here you are only changing the freestream velocity to include a vertical component while holding the collective (Thrust) constant.
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Post by Br. Marius on Apr 18, 2015 1:51:13 GMT
Agreed-I don't know how he got \(L_{NC}\) in his notes...I tried to figure it out last time and got no where. Only significant for torsion...alright. It would seem that the circulatory forces would be dramatically larger by intuition, so sounds good.
Alright, so we have a hovering helicopter, constant CT, and then it gets smacked by a sharp vertical gust. In retrospect, the effect that I spoke of wouldn't occur because it would be more like a cushion of air coming up, akin to ground effect. That's my hypothesis, anyhow...you've actually flown helicopters, so please let me know if something's up with my interpretation of helicopter performance.
So say that the gust comes and the helicopter hasn't yet reacted to it significantly. Now the gust has just reached the rotor from the bottom. It would subtract from up because it is approaching opposite the inflow, thus increasing the angle of attack of the blades. So the helicopter would get a sudden increase in lift (unless the blades stalled out) plus whatever vertical motion was induced by the gust's vertical drag on the helicopter itself...but we're not really concerned about that latter part here. Alternatively, if it approached from the top (did he establish a sign convention for that?) it would add to up and have the opposite effect. In order to remain at the same thrust coefficient in either case, the pilot would have to vary the collective pitch in response to the gust...but, on the other hand, the change in thrust coefficient can be computed because you can derive the change in AoA. I just don't feel right in leaving that \(\Delta C_T\) term as zero.
How does that sound to you?
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Post by matthorr on Apr 18, 2015 16:09:40 GMT
I've included \(\delta C_T\) in my equation - simple enough. I then didn't like having a \(\delta C_T\) hanging around where there are usually only thetas and lambdas, so I looked to BET for an expression for \(C_T\) in hover / axial flight, and then how the climb inflow is affected by vertical velocity (here \(V_g\). Put that all in and I got a nice expression that I think is closer to reality than ignoring \(\delta C_T\) since it is coupled to inflow and will change with a vertical gust. I made a few sign errors in the derivation (two in fact so they cancel in the end) and now I think I should change the orientation I assume for \(V_g\) to be downwards (same as \(U_P\)) to maintain consistency. So in the final equation, both of the signs inside the 4/3(...) should be reversed. From God's point of view (good tie in to religion...) it's as if He turned on a fan above the helicopter. From the pilot's point of view (assuming he is far above ground so he's not going to crash), it's as if he is all of a sudden in a climb condition, but trimmed for hover, so he would descend because he doesn't have enough collective pitch (effective inflow is increased, that subtracts from collective pitch in the equation for \(C_T\)). I think helicopter flying must be an Eastern religion because that sounds like something Buddha or Confucius would have come up with - "he who thinks he is climbing is actually descending." Scan 2015-4-18 0002.pdf (325.62 KB) |
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Post by Br. Marius on Apr 18, 2015 18:43:10 GMT
Haha...that's the funniest thing I've heard in awhile. Thanks for the explanation.
And that sounds good. Nice work!
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