Hey everyone, I need some help with blade element theory. I'm reading Richard von Mises' Theory of Flight and he makes a jump in the derivation of CT and CP that I don't understand. I'll post pictures of the diagram he uses and the equations he gives. Part of the book explains that calculations can be simplified by assuming that cos(gamma) is much larger than sin(gamma), gamma being the angle between the plane of rotation and the velocity vector, and by considering that CL is usually much larger than CD. However, in one of the pictures I'll attach, he inserts a seemingly random term in place of J2+(2pir/d)2. Is this just a good approximation?
Diagram:
Equations used for calculation, with substitution:
Equation for CP:
Equation for CT:
I don't have an answer to your direct question; but, based upon my experience with Von Mises solutions I suspect this is an accurate result of substitution and not an approximation. Have you tried to evaluate the basic equation terms conversions in this substitution?
Well, I know the equation with (sorry I'm having trouble with the equation editor) J2(1+cot2(gamma)) = J2/sin2(gamma) is an identity, and plugging in (1+cot2(gamma))/(sin2(gamma)) for the expression in brackets in equations 9 and 11 gives the equations in 12, so that much is correct. I just don't see how J2+(2pi r/d)2 equals either of the other two expressions.
I actually figured out that it is an approximation. For small values of gamma, sin(gamma) is approximately equal to tan(gamma), meaning for the purpose of calculation by hand (which was probably most common in 1959 when Theory of Flight was published) it is much simpler to use a value of V/2pi*rn for sin(gamma) than the actual value of V/sqrt((r*omega)^2 + V^2). I'll attach a picture of my solution, in case anyone else has trouble with it.
I had done a bit of the the same reductions before I sent my last post but could not make any connections with the results.
Good work!
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