# How could I calculate the torque due to lift and drag in the body frame?

I'm trying to figure out the torque due to aerodynamic forces on an aircraft confined to two dimensions (so no sideslip). Defining the windframe with basis vectors $$\hat{w_1}, \hat{w_2}$$ obtained by rotating by angle of attack $$\alpha$$ s.t. $$\hat{w_1}$$ coincides with the overall velocity in the body frame, I can apply lift $$F_L$$ and drag $$F_D$$ in $$-\hat{w_3}$$ and $$-\hat{w_1}$$, respectively. I guess adding together forces and calculating moments is best done in the body frame, so I transform using $$R = \begin{pmatrix} cos(\alpha) & -sin(\alpha)\\ sin(\alpha) & cos(\alpha) \end{pmatrix}$$ and end up with

$$[F_{aero}]_{body} = [-F_D cos(\alpha)+F_L sin(\alpha), 0, -F_D sin(\alpha)-F_L cos(\alpha)]$$.

So generally, along the body basis components I get some combination of lift and drag acting. However, where do these forces attack, and how can I find the exerted torque, i.e. what is their lever? Sorry if this is basic, any help is appreciated and please ask if I need to clarify anything. I was going to add a sketch but I can't see where to upload the file, apologies.

• In the edit option, there are icons. One of those icons is the add image option. It is square in shape and looks like a picture of a mountain and full moon.
– AJN
Commented Apr 15 at 14:51
• "...how can I find the exerted torque...". what you have described in the question is what is called a point mass model. Generally speaking, you cannot find any torques from such a model unless you already had some type of direct $\alpha \rightarrow C_M$ look up table / formulae with you. I would think that, the moment coefficient (or the lever) would be available from the same source from where you got $F_D$ and $F_L$. Use the edit option below the question to add more details.
– AJN
Commented Apr 15 at 14:53
• Thanks for your answer! I'm ultimately trying to find the hydrodynamic forces and moments on an underwater glider. Since there's not as much reading material I usually resort to reading about the analogous aerodynamics. Sorry if this creates confusion. I've done some reading and it seems there is indeed some coefficient vector that needs to be found. Say I had a design, how do I go about finding this coefficient vector. CFD? Commented Apr 15 at 19:23

There have been studies done on hydrofoils. There are similarities and differences between hydrofoils and airfoils. A major one is that water is incompressible and is assumed to be inviscid. As always, there are two hotly debated explanations for the lift created by the hydrofoil.

Bernoulli:

$$P_0=P_1 + ½\rho v_1² = constant$$

We can disregard the term $$\rho g* y_1$$, because y1 is small.

• Po Stagnation Pressure [Pa] or [lbf/ft2]
• P Pressure [Pa] or [lbf/ft2]
• $$\rho$$ Density [kg/m3] or [lbf/ft3]
• V Velocity [m/s] or [ft/s]
• g Gravitational Constant [m/s2] or [ft/s2]
• y Height [m] or [ft]

.

Euler:

$$d(p+\rho g *y)/dn = \rho v²/R$$

• P Pressure [Pa] or [lbf/ft2]
• $$rho$$ Density [kg/m3] or [lbf/ft3]
• V Velocity [m/s] or [ft/s]
• g Gravitational Constant [m/s2] or [ft/s2]
• y Height [m] or [ft]
• n Vector in Radial Direction ---
• R Radius of Curvature of Streamline [m] or [ft]

'

Angle of attack: By positioning the hydrofoil at an angle with respect to the stream, and by attaching trim tabs to its rear edge the lift can be adjusted.

.

Here is an MIT course on the hydrofoils link. souce