Aerodynamics and Design for Ultra-Low Reynolds Number Flight

Chapter 6
The governing equation relates the rate of twist to the applied torque subject to the
boundary conditions that Q(tip) = 0 and θ(0)=0:
The polar moment of inertia for a thin flat plate can be expressed as:
What remains is to derive an expression for the torque applied to spanwise differential
element. The model used in deriving the toque expression is depicted in Figure 6.24. A
differential element of an inclined blade section is displaced above or below the
rotational plane by some distance h. This element having mass dm experiences the
centripetal acceleration ω . The resulting force vector has no component in
2 rˆ dmω
the y direction and a component in the x direction proportional to Sin(β). The moment
about the structural axis due to dm can be expressed as:
This is then expressed in terms of c, t/c, ρ material, R, and ζ, and integrated across the
chord to yield:
This would be the torque without any torsional deflection, and would provide a first
order estimate, but the actual torque will be a function of the built incidence and the
torsional deflection, taking the final form:
120
J( r)
Q( r)
dθ
( r)
dR
tip
1
= -------------- QR ( ) dR
GJ( r)
cr ( )tr ( ) 3
= --------------------- =
3
∫
r
1
-- cr ( )
3
4 t
- ( ⎛ r ⎞ )
c ⎠ ⎝ 3
dQ( r)
mω 2 r ˆ = – d hsin( β)
1
=
–
12
-----ρ mat · t
- c
c
4 ω 2
cos(
ζ)
sin(
ζ)
dR
2 rˆ
(6.1)
(6.2)
(6.3)
(6.4)