Beam Model

The beam model used is the single-celled composite beam model of Rehnfield [3]. While the model is quite simple; use of it is somewhat dubious.

First of all, the blade I am designing is double-celled. Only the trailing part of the blade (the spar and the skin behind the spar) are modeled structurally. The skin ahead of the spar produces quite a bit of torsional stiffness; however, it is expected that careful design using coupling behavior of composites will offset the added stiffness.

Second, Rehnfield's model only applies to thin-walled beams. The spar, however, is not thin walled. In fact, the spar about as thick as the thinner airfoils themselves.

Despite the doubtfulness of the model, it offers the simplest analysis, and so is used for the very early preliminary design.

Rehnfield's model relates seven generalized forces to seven generalized strains. The force vector is

$$\vec F = [\ N_x\ Q_y\ Q_z\ M_x\ M_y\ M_z\ Q_\omega\ ]^T.$$

$N_x$ represents an axial force, $Q_y$ and $Q_z$ shear forces, $M_x$twisting moment, $M_y$ and $M_z$ bending moments, and $Q_\omega$ a generalized warping force. The strain vector is

$$\vec\epsilon = [\ U,_x\ \gamma_{xy}\ \gamma{xz}\ \phi,_x\ \beta_y,_x\ \beta_z,_x\ \phi,_{xx}\ ]^T.$$

$U$ is the axial extension, $\gamma_{xy}$ and $\gamma_{xz}$ are shear strains, $\phi$ is the angle of twist, and $\beta_x$ and $\beta_y$ are bending angles.

$\vec F$ and $\vec u$ are linearly related by a stiffness matrix:

$$\vec F = [C]\vec u.$$

The components of the stiffness matrix are functions of the beam geometry, including ply layup. Equations for these components are given in Equations 21 of Reference [3].

For the present design, the components of $[C]$ are calculated piecewise linearly, with the three pieces being the lower skin, the spar, and the upper skin. The Appendix lists a Mathematica script that determines formulas for the components of $[C]$ for a single piecewise-linear member.

Once $[C]$ is obtained, the equation $\vec F = [C]\vec u$ can be solved for $\vec u$. The fourth component of $\vec u$, $\phi,_x$ can be integrated to determine the twist of the blade.