User:Eas4200c.f08.blue.a/Lecture 16

Continuing the discussion on multicell torsion:

The two important equations for single cell torsion are

$$T=2*q*A$$

and

$$\theta=\frac{1}{2(G)(A)}\oint\frac{q}{t}dl$$

Which is the torque and angle of twist respectively of the cell.

when moving to a multicell view in which cell $$i=1,....,n_{cell}$$

Torque on the multicell system now becomes equal to the sum of all the individual cell torques.

$$T=2*\sum{q_i*A_i}$$

in which

$$q_i$$ is the shear flow in cell i

$$A_i$$ is the "average" area in cell i

so therefore $$T_i=2*q_i*A_i$$

so the torque on the for the entire multicell system would then be the sum of all the $$T_i$$

$$T=\sum_{n=1}^{n_c}T_i$$

the same approach is used for the angle of twist $$\theta$$

in which

$$\theta= \theta_1,.....,\theta_i$$

Therefore

$$\theta_i=\frac{1}{2*G_i*A_i}\oint\frac{q_i}{t_i}dl$$

in which

$$G_i$$ is the shear modulus of cell i

and

$$t_i$$ is the thickness of cell i

so therefore the angl$$e$$ of twist for the entire multicell system would then be the sum of all the $$\theta_i$$

$$\theta=\sum_{n=1}^{n_c}\theta_i$$

Example



$$t_1=.008m$$

$$t_2=t_3=.01m$$

$$a=4m$$

$$b=2m$$

therefore the area will equal the sum of the semicircle and triangle which make up the overall airfoil shape.

$$A=\frac{\pi*(\frac{b}{a})^2}{2}+\frac{1}{2}*b*a=5.5708m $$

and

$$T=2*q*A$$

so therefore

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

in which T is the variable

so the twist angle will equal

$$\theta=\frac{1}{2*G*A}\sum_{j=1}^{3}\frac{q_j*l_j}{t_j}$$

in which l is the length of the path.