Rotor Dynamics Quantification HELP

[Bad-Wolf]

Hi, I'm a UT Mech E student about to graduate who's been tooling around with with some helicopter dynamics books for home machined RC contrarotating helicopter fun and I had some questions for any aerospace pros lurking around here. Being ME I know few aerospace guys so I've got some burning questions. I'm machining everything other than the glow motors (.40 in^2 that deliver .92kW @ 15,000rpm) and the RC electronics gear of course. I've got a couple courses in fluid mechanics under my belt and just started tearing up a helicopter dynamics book a few weeks ago but it is all leaving quite a bit to be desired. I am trying to quantify all this theory and gain a bit of intuition as to the working of these ungainly equations I've been using, but I think I'm mixing everything up or something.

the following is mostly concerning hover
I understand that the in the case of the rotor:
-the rotor is like an airplane wing in rotation (over simplified of course)
-thus we need to be concerned with the difference in velocity as we move out to the ends of the radius of the "wing"
-additionally because the blade area is essentially in a fixed position in hover there is something called induced velocity (i hope that is right) which is the velocity imparted to the air by the rotor to essentially move it out of the way (from momentum theory?)

Now if I were to use Blade element theory and momentum theory ( and the formulas associated ) what kind of approximation would the numbers generate compared to the real conditions for something RC sized? Like what percent error compared with reality. I am trying to figure out what torque increase at rpm cost is best given a feasible amount of lift.

How much lift can I actually generate with this motor?
It gets all jarbled up when I look at the math because Lift and Drag increase as rpm (disregarding pitch at the moment). Drag is the source of my torque loading. Thus I need to strike a balance between lift (thus rpm, radius, pitch; disregarding camber) and torque consumption.

All the theory makes sense; I think I just need some way to systematically look at the design of this rotor given my one given; glow motor: .92kW, 15,000rpm A bit of guidance would be much appreciated. Guidance being the key after that jumble I wrote up there lol. I guess it's just hard using theory I've literally just acquired and haven't gone through the whole solve lots of problems for practice student rigamorol and frustrated because this kind of research and immediate application is what I feel like I should have been taught in addition to all this theory in school.

I'll post more tomorrow and look forward to responses as it's late and I got early class :P

Edit: I'm just really interested in the best way to solve this problem through personal study etc rather than taking a class in it. Learning how to do research properly myself and all that.

 

[Bad-Wolf]

I need to know if anybody knows a method for approximating a mean coefficient of drag on a rotor using the coefficient of lift and the Reynold's number at 75% of the radius if given that I know the theoretical values for for all that? My book talks about it but doesn't really show me how to do it. It's the last thing I need to figure out approximate values of the power required for my theoretical rotor :(

Edit: Oh ya; I answered just about all my questions and figured out everything that was bugging me. Now I just need a nifty message for a numerical approximation like I ask above.

 

[Bad-Wolf]

wow, really, no responses ? :(

 

[Cyrus]

the following is mostly concerning hover
I understand that the in the case of the rotor:
-the rotor is like an airplane wing in rotation (over simplified of course)

I wouldn't state it like that. You can think about it that way, but its best not to state it like that for certain reasons on how the flow behaves.

-thus we need to be concerned with the difference in velocity as we move out to the ends of the radius of the "wing"

Yes: Replace "wing" with "rotor blade". The velocity distribution (in hover) is linear with radial distance.

-additionally because the blade area is essentially in a fixed position in hover there is something called induced velocity (i hope that is right) which is the velocity imparted to the air by the rotor to essentially move it out of the way (from momentum theory?)

I don't understand waht you mean by "blade area si esentially in a fixed position". This doesn't really mean anything, and I would avoid saying this. Induced velocity is a result of having to push the flow below the rotor, not because of a 'fixed area'.

Now if I were to use Blade element theory and momentum theory ( and the formulas associated ) what kind of approximation would the numbers generate compared to the real conditions for something RC sized? Like what percent error compared with reality. I am trying to figure out what torque increase at rpm cost is best given a feasible amount of lift.

The accuracy doesn't matter on the 'size', but on matching the Reynolds number of the airfoil data you have for your blades. If you are using 1-d flow for momentum theory, and you have a program like x-foil or the like to generate drag polars at the reynolds number you expect to see on the RC helicopter, you should be pretty close.

How much lift can I actually generate with this motor?

Just use momentum theory to calculate the power required to hover at a given weight that is 95% of your maximum engine power. Note, you can't use 100% power in hover or you have no power available to climb.

It gets all jarbled up when I look at the math because Lift and Drag increase as rpm (disregarding pitch at the moment). Drag is the source of my torque loading. Thus I need to strike a balance between lift (thus rpm, radius, pitch; disregarding camber) and torque consumption.

Not really. See above.

BTW: Which book are you using?

 

[Bad-Wolf]

Thanks for the response. Most of what you said I just miss-worded and gotten hip with the jargon since. However, I think you can in fact help me. The book I used after a good sample of 5 different helicopter dynamics books was Helicopter Theory by Wayne Johnson.

By fixed area at the time I meant that the rotor is not in translational motion only rotating. The rotor plane is not moving.

The semantics between calling it a rotating wing versus a rotor is pretty meh but I'll endeavor to use the correct phrasing. I figured the (over simplified) qualifier would be enough for anybody not to reiterate something I was already aware of. This is my fault; I will endeavor to be clearer in all my future engineering writings.

Now then, for the actual question. I'm not interested in calculating some ideal power consumption from the use of momentum theory. I need the inclusion of blade element theory because I am using this data to also design my blades without experimentation. I've since done what you said but still need to know what gear ratio I need to use to achieve the optimal torque to rotational velocity. If I can calculate the drag I'll know the torque required for the same amount of lift. So my question is this.

Using the lifting line theory that blade element utilizes and the induced velocity from momentum theory I've already got good equations for coefficients of lift and lift itself that are quite accurate in addition to that for drag. However, I do not have a good method for finding the coefficient of drag on my rotor. The book says that I can calculate a good mean coefficient of drag by considering the radius to be at 75% and using the reynolds and mach numbers at that point. If you know how this is done I would appreciate it considerably. As far as a program for doing this; the entire purpose of this exercise would be lost...

 

[Cyrus]

Thanks for the response. Most of what you said I just miss-worded and gotten hip with the jargon since. However, I think you can in fact help me. The book I used after a good sample of 5 different helicopter dynamics books was Helicopter Theory by Wayne Johnson.

That's the real-deal book for the subject. I use Leishman's book "Principles of Helicopter Aerodynamics". So I could reference you to pages in that book if you can get a copy of it.

By fixed area at the time I meant that the rotor is not in translational motion only rotating. The rotor plane is not moving.

Ok, I wouldn't say that because it doesn't make any sense from a helicopter dynamics perspective.

The semantics between calling it a rotating wing versus a rotor is pretty meh but I'll endeavor to use the correct phrasing. I figured the (over simplified) qualifier would be enough for anybody not to reiterate something I was already aware of. This is my fault; I will endeavor to be clearer in all my future engineering writings.

The reason why I don't like seeing it called a wing is because unlike a wing the flow physics is different. You get things like the vortex ring state, high speed impact noise, and blade-vortex interactions that you don't get on a wing. The analogy is good for explaining it to a layperson, but being that you are an engineer I would not call it a wing.

Now then, for the actual question. I'm not interested in calculating some ideal power consumption from the use of momentum theory. I need the inclusion of blade element theory because I am using this data to also design my blades without experimentation. I've since done what you said but still need to know what gear ratio I need to use to achieve the optimal torque to rotational velocity. If I can calculate the drag I'll know the torque required for the same amount of lift. So my question is this.

Using the lifting line theory that blade element utilizes and the induced velocity from momentum theory

-In hover.

I've already got good equations for coefficients of lift and lift itself that are quite accurate in addition to that for drag. However, I do not have a good method for finding the coefficient of drag on my rotor.

I don't understand this question because a rotor generates thrust and you calculate $$C_T$$ using BEMT. The drag coefficient at each blade station comes into play when you calculate the power (or equivalently torque) requirements.

If you could scan the pages you're talking about that would help. Right now I'm a bit rusty as I did this a while ago, but I'll pick it up again as I help you so double check what I say with what's in your book.

 

[Bad-Wolf]

A bit of a remission, but I return with more questions and clarifications. To start I shall include some equations for questioning. Also, all of my concerns are for rotors in hover.(1) $$C_{T}=\int^{1}_{0}\frac{\sigma}{2}r^{2}c_{l}dr$$

(2)$$\sigma=\frac{Nc}{\pi R}$$
this assumes constant chord (i need to revise this for blade taper)
N: Number of blades
c: chord
R: radius

(3)$$c_{l}=a\alpha$$
a: rotor slope of leading edge
$$\alpha : angle of attack$$

(4)$$C_{p}=\int \lambda dC_{T}+\int^{1}_{0}\frac{\sigma}{2}r^{3}c_{d}dr$$

(5)$$\lambda=\frac{\sigma a}{16}[\sqrt{1+\frac{32}{\sigma a}\theta r}-1]$$

These coefficients are all well and good but I need to see actual numbers for lift and drag so I have the following which will require integrations:

$$L\cong 1/2\rho u_{t}^{2} ca(\theta-u_{P}/u_{T})$$
$$D\cong 1/2\rho u_{t}^{2}cc_{d}$$

$$dT\cong NLdr$$
$$dQ\cong N(L\phi+D)rdr$$

There is a linear relationship between the coefficient of lift and drag which I can use for the drag coefficient in this case but I don't know how to find it though I could probably figure it out. Sooooo...

If I integrate the two canned equations which hold quite a few assumptions that should hold true enough for a .75m radius disk plane and a .92KW motor can I expect to get reasonably accurate values of thrust and rotor shaft torque? I need to design my rotor so I can then know how much torque and rpms my gear box is going to need to spit out. I realize that the power loss due to the tip vortices is not included. Could I just assume that whatever values I get from the equations needs to be reduced by something like 10% to account for vorticial/tip losses? Those big equations for Cp and Ct up at the top of my post are nice for maximizing performance and all but if I have no clue how much actual thrust is generated and how much power is required they are useless to me. Maybe I am not seeing something as I would really like to use those much more inclusive coefficients in calculating my thrust and power consumption.

That book you mentioned is at our engineering library in like 4 different publishing years and I think 2 of them are available. If you can point me to where I can find some math that takes more of this into account for calculating thrust and power required for a given rotor design perhaps utilizing vortex theory as well or something. It is very annoying working with incomplete theory. I would like to be able to calculate accurately just from theory the thrust and power required for a predefined rotor.

Once I've got a set of relationships that are accurate to say 5% of the actual values I'd like to setup a program so I can dump in lots of different values and play with the equations to get a good feel for how each value effects my thrust and power. Ultimately my means of actually changing thrust on the thing itself would be varying the pitch. Thus I would like to see some numbers I get when I mess with the values like:
angle of twist (root incidence angle and tip incidence angle)
chord (taper ratio possibly if not constant chord)
airfoil slope (camber)
radius
number of blades
rotational velocity
torque

Edit*: Found the following equation utilizing the coefficient of thrust online. Using the equation for the coefficient of thrust up above do you think this equation will produce a fairly accurate value for thrust? Could you possibly take a look in leishman for a good equation for the P Power or Torque Q?

$$T=C_{T}(\rho A(\Omega R)^2)$$

 

[Cyrus]

Hover I can def. help you with. Take your time and write out what you got and I'll help where needed. If I don't know I'll just make up an answer because you'll be none the wiser :tongue2:

 

[Bad-Wolf]

If you make up numbers it will be blatantly apparent unless the numbers are very close to what I expect but untruthfully accurate given the theory you used.

Just by assuming some nominal value for the figure of merit, I can make some assumptions as to the power required to generate a given amount of thrust.

Grrr, I just need to get a copy of leishman; looking through pieces of his book online and it seems to be much better written : /

Well, after frustrating myself again I've decided to go hunt down a copy of leishman and read quite a bit more.

 

[Cyrus]

Let me know if you have any questions after getting Leishmans book. I was going to type a how to thread on rotor aerodynamics, but it's too much work on my part. It's easier for me to simply point others to lieshmans book because, ultimately, that's what I'm retyping anyways. Then you can just tell me what page your stuck on and I can turn to it. (It's a hell of a lot easier for both of us).