# How to locate the coordinates of the Center of Gravity of an airfoil?

I am working on the inertial effects on a variable pitch propeller and i need to define the location of the center of gravity on the airfoil to calculate moments of Inertia and angular momentum. What is the best way to do it?

• The same way you locate the center of gravity of any object. If it is very simple you do summation. If it is a curve defined by a formula you do integration. If it is anything else you use numerical methods. – kevin Aug 17 '17 at 20:20
• You can get reasonably accurate results by cutting a piece of plywood with the form of the profile, suspending it at three different points, and then mark the prolongation of the suspension cord on the profile. The intersection point of the lines (or the center of the small triangle, if one is thus generated) marks the c.g. – xxavier Aug 18 '17 at 8:37
• Perhaps a question for you would be this -- are you trying to determine the CG location analytically and experimentally? It sounds like you're trying to design the prop on paper and do some design studies...so my second question is this: what are you trying to do? – Marius Aug 21 '17 at 15:42
• @Marius Yes, actually i am designing a propeller blade with the pitch axis placed in an offset distance from the aerodynamic centre or CG. So i am trying to calculate the pitching moments due to mass/Inertial effects, and i need parallel axis (Steiner's) to make the calculations which are made with the CG location as a reference. – george Aug 21 '17 at 19:05
• Oh ok. I just had to do something similar for computing aeroelastic effects on a wing. The only way I saw how to do it was to make a little calculator that would compute the cross-sectional properties for me. Essentially, it's the same of @kevin's comment -- just integrate to find the cross-sectional properties given some desired skin thickness. I don't know if there is a "best" way to do what you want rather than just creating "a tool that does what you want" based on the design of your cross sections. What do your cross sections look like, and what airfoil(s) are you using? – Marius Aug 21 '17 at 19:16

## 1 Answer

To compute the inertial properties of your blade, you need a couple of things: a way to compute inertial and elastic properties along the blade, a blade layout, and some idea of the expected load you want this to carry.

Starting with elastic properties--this can be either really straightforward or VERY tricky, especially when you start throwing composites around. That said, this is true primarily because composites can be tailored into having just about whatever material properties you want...but the analysis of composites for essentially what you're doing was the PhD dissertations of two grad students I know. The one is developing an analysis tool for aeroelastically tailored blades (also mentioned here, in a study by Sandia National Labs: http://energy.sandia.gov/wp-content/gallery/uploads/SAND2001-1303A.pdf), which sounds like what you're trying to do. In short, these blades take advantage of strategically tailored material properties in order to achieve variations in blade twist purely based on a blade's operating condition. If you are going to develop your own composites design code (or have one!), I think this is a feasible goal and I'd recommend you to "Analysis and Performance of Fiber Composites" by Agarwal, Broutman, and Chandrashekhara for an intro to composite analysis. Based on that text, you could develop an analytic design code that might get you into the ballpark, and maybe start developing some finite element models to go beyond that. It requires a fair amount of setup, and, since I don't think it was the main thrust of your question, I'm not going to write it out.

I think it suffices to say that back-of-the-envelope estimates for composite materials are hard to come by. That said, it can be a rough approximation to treat composites as "black aluminum", or as an isotropic material with the bulk mechanical properties of the material (though it nullifies the aeroelastic tailoring you're shooting for). Note that this requires you to have a symmetric layup schedule, as described in this short course from the US Naval Academy (https://www.usna.edu/Users/mecheng/pjoyce/composites/Short_Course_2003/7_PAX_Short_Course_Laminate-Orientation-Code.pdf). This isn't the main thrust of your question, however, so I'm not going to delve into it further...it's just a head's up since composites can be more trouble then they are worth, especially if you have access to more traditional machining resources that you can use to make blades out of materials that are easier to shape (plastics, etc.). The FAA has a nice guidebook too, and, depending on how fluent you are with this stuff, it may be of interest: https://www.faa.gov/regulations_policies/handbooks_manuals/aircraft/amt_airframe_handbook/media/ama_Ch07.pdf.

If you're not shooting for aeroelastic tailoring or your analysis, for whatever mission you're planning for doesn't show that you get a terrible performance gain, it may be worth it to just try and make a variable pitch mechanism instead.

Alright -- on to inertial properties. If you're looking to compute the blade's dynamic stability, you'll need inertial properties around multiple axes of rotation to account for the 3 modes of blade motion: blade flapping (i.e., motion of the blade out of plane with the rotation of the propeller), pitching (or twisting), and lagging (motion of the blade in the plane of rotation that occurs in addition to rotation of the blade itself). Blade stability is related to the coupling of these three modes. For an application I had (wing of constant cross section, isotropic material), I assumed the following cross section, including a quarter-chord spar and skin.

To solve for the inertial properties of the section (so that I could do a simplified aeroelastic stability analysis), I moved point by point along the airfoil surface. For example, take the picture below. It's a zoom in on the upper surface of the previous image.

To find the signed area of a convex polygon, in this case the quadrilateral defined by P1, P2, P3, and P4, the equation is (where x is in the chordwise direction, z is in the vertical direction, and y is the spanwise direction along the blade):

$A=0.5\sum_{i=0}^{n-1}(x_iz_{i+1}-x_{i+1}z_i)= 0.5\left[((x_{P1}\cdot z_{P2})-(x_{P2}\cdot z_{P1})) + ((x_{P2}\cdot z_{P3})-(x_{P3}\cdot z_{P2})) + ...\right]$

...and so on. Similarly, the centroid can be determined by

$x_c = \frac{1}{6A}\sum_{i=1}^{n-1}(x_i+x_{i+1})(x_iz_{i+1}-x_{i+1}z_i)$

(formulas are most legibly given at https://en.wikipedia.org/wiki/Polygon#Area_and_centroid). Just use the parallel axis theorem (essentially just weighted average of the centroids and areas) to get all these individual pieces to yield the sectional area and centroid (which, for an isotropic material at least, is the location of your neutral axis).

Moving on the second moment of area (which you'll need for computing stability), it's a similar formula (found at https://en.wikipedia.org/wiki/Second_moment_of_area#Any_polygon):

$I_{xx}=\frac{1}{12}\sum_{i=1}^n(z_i^2+z_iz_{i+1}+z_{i+1}^2)(x_iz_{i+1}-x_{i+1}z_i)$

($I_{zz}$...is similar -- just swap x and z). Also:

$I_{xz}=\frac{1}{24}\sum_{i=1}^n(x_iz_{i+1}+2x_iz_i+2x_{i+1}z_{i+1}+x_{i+1}z_i)(x_iz_{i+1}-x_{i+1}z_i)$

Apply the parallel axis theorem to get all these individual moments to combine into a sectional property.

Once you have those implemented, the question is just what area do you want to integrate (i.e., is the blade solid, how large is your LE weight, etc.), and what are the elastic properties of those sections. Then just apply the above. The out of plane moment of inertia is the sum of the in-plane moments (i.e., $J = (I_{xx}+I_{zz})$).

Finally, as to actually computing dyanamic/static stability, I have a reference text...but it's a set of course notes and I'm not sure the best way to post it as the method is quite involved. My apologies, but I'm coming up a little short on a good reference for this and may update this question later if I do find something good, however, in the interim, I'd advise you to look at helicopter dynamics texts, which will be a source of material for performance analysis computations too. One source I could find online is by Bramlette and seems to have the basics well covered, but doesn't venture into rotor stability analysis: http://airspot.ru/book/file/63/bramwell_helicopter_dynamics.pdf

• I have read your explanation about the Inertial properties and stability, however i am a bit confused on the axes chosen for the moments and products of Inertia. On the "zoom-in wing skin" figure we have z vs x axis while on moments of inertia working on x and y axis. Also you mentioned the parallel axis theorem but as i understood, you meant to use it along the span of the blade to find the weighted C.G. What about parallel axis theorem on pitch axis - and C.G axis? Finally you talking about centroid instead of C.G. Is that because you assumed a constant mass distribution right? – george Aug 23 '17 at 13:24
• My apologies about the xz vs xy mixup -- I'm so used to using xz ordinates for airfoils, but the formula was stated in xy instead. I've changed variables in the formulas to be consistent. As for the parallel axis theorem -- you can use it for more than CG. I'll write it out in an edit tonight. And yes, I do just find the centroid of the body for the CG because I assume a constant density, but it's not too different to find the CG of a nonhomogeneous cross section...I'll write that in too. – Marius Aug 23 '17 at 13:54
• This is a sketch i have made based on my understanding on how to use Inertial effects i.stack.imgur.com/qQAiw.jpg I think there are some similarities with your perspective however i am still confused on which axes i need consider. What about the 3rd axis and therefore moments of inertia related to that axis? and where to define the reference frame coordinate system? Those are the formulas i 've ended up with: dm: ρdxdydr; Offset Pitch Axis - x; Offset longitudinal Axis - y; Force: (ρdxdydr)*rω^2 ; Component along the x-axis: (x/r)*Force; Pitch Moment: y*(x/r)*Force – george Aug 23 '17 at 14:22
• The inertial and mass effects that you should also consider, in addition to what you've already mentioned include blade flapping (i.e., motion of the blade out of plane with the rotation of the propeller), and pitching (or twisting). Blade stability is related to the coupling of these two modes of blade motion in addition to the one you've already described. You may want to ask a full, second question about all that stuff. It's too much for here. – Marius Aug 24 '17 at 1:42
• i have uploaded a new question here aviation.stackexchange.com/questions/43173/… Thanks for the assistance anyway – george Aug 24 '17 at 9:36