User:Eas4200c.f08.WIKI.E/hw3

Friday Octopber 8th
K) 1) Multicelled Section

For cells where $$i = 1, ... , n_{cell}$$



$$\tau = 2\sum_{i=1}^{N_c}{q_i \bar{A_i}} $$

$$q_i \;$$ = Shear flow in cell i.

$$\bar{A_i}$$ = "average" area in cell i.

Define: $$T_i = 2q_i\bar{A_i}$$ torque generated by one cell.

$$\tau = 2\sum_{i=1}^{N_c}{q_i \bar{A_i}} $$

$$T_i = \sum_{i=1}^{N_c}{T_i} $$

2) Shape of airfoil is "rigid" in the plane $$(X,Y)$$ however it can warp out of the plane.

$$\theta = \theta _1 = \theta _2= ...=\theta _{N_c}$$

The twist angle equation applied to cell i.

$$\theta_i =\frac{1}{2 G_i \bar{A_i}}\oint_{}^{}{\frac{q_i}{t_i}ds}$$

$$G_i\;$$ = Shear Modulus of cell i

$$t_i(s)\;$$ = thickness of walls on cell i. This is a linear curve coordinate along the cell wall.

''' Pb 1.1: Rectangular single cell section with modification. '''

Now more general with a single cell section:

$$t_i=0.08 \; m$$

$$t_2= t_3=0.01 \; m$$



$$\bar{A}=\frac{1}{2} \pi (\frac{b}{2})^2 +\frac{1}{2}ba= 5.5708$$

Shear flow comes from the torque equation: $$T=2q\bar{A}$$

$$q= \frac{T}{2\bar{A}}$$

Twist angle: $$\theta_i =\frac{1}{2 G_i \bar{A_i}}\sum_{j=1}^{3}{\frac{q_jl_j}{t_j}}$$ $$j=1,2,3$$ index for the segment number.

There is only one cell for $$\theta_i\;$$ so it reduces to $$\theta\;$$ in the equation above. You would normally have two indexes for multicelled section $$(i,j)\;$$.

$$\theta =\frac{1}{2 G \bar{A}}\sum_{j=1}^{3}{\frac{q_jl_j}{t_j}}$$

Note: $$\int$$ is and elongated S, standing for continuous summation. $$\sum{}$$ is a discrete sum.