User:EAS4200C.Fall08.AERO.Watlington.VG

10/3/08 Lecture 17

Roadmap continued...

K. Multi-cell section (cell i=1,...$$n_c$$):

K1. The total torque T for a multi-cell section is defined as:

$$T=2\sum_{i=1}^{n_c}{q_i\bar{A}_i}$$

where $$q_i$$ is the shear flow in cell i, and $$\bar{A}_i$$ is the average area in cell i.

Justification for Torque equation

Define $$T_i=2q_i\bar{A}_i$$

as the torque generated by one cell. It then follows that T is the sum of all individual cell torques.

K2. The shape of air foil is rigid in the (y,z) plane (however, it can warp in and out of the plan in the x-direction.)

This rigidity leads to the following relation about the angles of rotation for each cell:

$$\theta =\theta _1=\theta _n$$

and $$\theta =\frac{1}{2G_i\bar{A}_i}\oint_{}^{}{\frac{q_i}{t_i}ds}$$

where $$G_i$$ is the shear modulus of the cell and $$t_i$$ is the thickness of each segment. $$t_i$$ can also be a function of $$s$$, the curvilinear coordinate along the cell wall.

Example

Now, let us refer back to Problem 1.1 to demonstrate the above equations in a single cell section.

(picture of cell goes here)

First, calculate the area: $$\bar{A}=\frac{1}{2}\pi(\frac{b}{2})^2 + \frac{1}{2}(b*a)$$ = $$\bar{A}=\frac{1}{2}\pi(\frac{2}{2})^2 + \frac{1}{2}(2*4) = 5.57$$

Then, Shear flow: $$T= 2q\bar{A}$$ => $$q=\frac{T}{2\bar{A}}$$

Finally Twist Angle: $$\theta =\frac{1}{2}G\bar{A}\sum_{j=1}^{n_c} q_j\frac{l_j}{t_j}$$, where $$j$$ is the index for segment number within the cell.

NOTE:

integral sign: $$\int$$, standing for summation (continuous)

summation: $$\sum{}$$, (discrete) sum

This method is derived from the Riemann sum, which is computed by dividing a body into smaller rectangles and adding their areas together to arrive at a total area for the body.