Long time lurker, first time poster. I am working on a custom design (purely thought experiment at the moment; more details when it's more mature), and am having difficulties with my first-level approximation of wing moment. Assume a wing with highly variable properties (coefficient of lift, coefficient of moment, chord, etc.) across the span, and non-constant sweep. Actively ignore interference effects and spanwise flow; this is a 2D approximation only. The total lift of such a wing can be found by the lift equation (assuming symmetrical wings):
2 * integral from 0 to span/2 of ( Cl(x) * Chord(x) * 0.5 * rho * v^2 )
where Cl(x) is the coefficient of lift at spanwise position x, and Chord(x) is the local chord at spanwise position x. Dimensional analysis suggests this formula is correct, and it matches up with known formulas for simple cases. For stable flight, it's easy enough to solve this for lift == weight for interesting cases.
Where I'm hitting problems is finding the integral form of the moment equation. Assuming Cm0(x) is the coefficient of moment of the airfoil at spanwise position x and Arm(x) is the longitudinal distance from the center of lift at spanwise position x to the center of gravity (negative for center of lift behind center of gravity), then I believe we have:
Cm(x) = Cm0(x) + Arm(x) Cl(x) / Chord(x)
to give the local coefficient of moment... but I may already be off in the weeds here, as I'm not fully convinced we use the local chord (rather than a reference chord) in this formula.
By analogy to the lift equation, we can then try to get the total moment from the above... but here I'm stumped. Dimensional analysis suggests one of:
2 * integral from 0 to span/2 of ( Cm(x) * Chord(x)^2 * 0.5 * rho * v^2 )
or
2 * integral from 0 to span/2 of ( Cm(x) * Chord(x) * Arm(x) * 0.5 * rho * v^2 )
The former seems incorrect, though, as the Cm0(x) term never has its distance from the center of gravity factored in. The latter seems incorrect in that the Arm(x)^2 term that develops for the contribution of lift to moment prevents this portion of moment from ever being negative.
Given this formula for moment, of course, I'd want to solve for trim (moment == 0) and static longitudinal stability (derivative of moment with regards to coefficient of lift == 0 at neutral point).
So at what point am I going wrong here?
2 * integral from 0 to span/2 of ( Cl(x) * Chord(x) * 0.5 * rho * v^2 )
where Cl(x) is the coefficient of lift at spanwise position x, and Chord(x) is the local chord at spanwise position x. Dimensional analysis suggests this formula is correct, and it matches up with known formulas for simple cases. For stable flight, it's easy enough to solve this for lift == weight for interesting cases.
Where I'm hitting problems is finding the integral form of the moment equation. Assuming Cm0(x) is the coefficient of moment of the airfoil at spanwise position x and Arm(x) is the longitudinal distance from the center of lift at spanwise position x to the center of gravity (negative for center of lift behind center of gravity), then I believe we have:
Cm(x) = Cm0(x) + Arm(x) Cl(x) / Chord(x)
to give the local coefficient of moment... but I may already be off in the weeds here, as I'm not fully convinced we use the local chord (rather than a reference chord) in this formula.
By analogy to the lift equation, we can then try to get the total moment from the above... but here I'm stumped. Dimensional analysis suggests one of:
2 * integral from 0 to span/2 of ( Cm(x) * Chord(x)^2 * 0.5 * rho * v^2 )
or
2 * integral from 0 to span/2 of ( Cm(x) * Chord(x) * Arm(x) * 0.5 * rho * v^2 )
The former seems incorrect, though, as the Cm0(x) term never has its distance from the center of gravity factored in. The latter seems incorrect in that the Arm(x)^2 term that develops for the contribution of lift to moment prevents this portion of moment from ever being negative.
Given this formula for moment, of course, I'd want to solve for trim (moment == 0) and static longitudinal stability (derivative of moment with regards to coefficient of lift == 0 at neutral point).
So at what point am I going wrong here?
