User:Eas4200c.f08.aeris.guan/hw7

Matlab HW7: Determining the True Shear Flow in Different-Celled NACA Airfoils
The problem statement in this homework was to determine the true shear flows of two different cases of the NACA airfoil subject to shear forces. Part 1 was to determine these shear flows in a single-cell airfoil and part 2 was to determine these shear flows in a double-cell airfoil. The airfoils in both cases have four stringers: 2 positioned at a quarter of the chord length and 2 positioned at three quarters of the chord length. In each case, the problem was solved using superposition of the shear flows in a case analyzed without stringers and a case with stringers. The equation of moment and the compatibility equation were also used to find the true shear flows. The code in which was developed to solve this problem is shown in the collapsible boxes below. For part 1, the code starts out with finding the values ky, kz, and kyz by using the moment of inertias obtained by previous homework assignments. Then, it goes about solving the P2 problem as posed in lecture. The trailing edge is the portion of the skin that is designated to be cut. Here, Qy and Qz are calculated at each stringer node and the equilibrium equations of the shear flows at each stringer are written. This solves for all the q tilda's in P2. After these shear flows are obtained, it is necessary to find a the shear flow of the two cells (q_11 and q_21) solving for the P1 problem as posed in lecture. The indices in the notation are described as follows: the first subscript is the cell number, and the second subscript is the part of the homework the shear flow corresponds to. To do this, the moment equation is used and the moment was taken about point D. To find the moments contributed by the shear flows found in P2, the code was written to sweep from stringer to stringer that had a non-zero shear flow. At each sweep, an infinitesimal length of the skin was calculated and multiplied by the shear flow through that skin portion to obtain a force. Once that force was determined, the perpendicular distance from point D to the line of action of the force was obtained and then multiplied to the force to get an infinitesimally small moment. All the moments were added together to get the moment produced by each shear flow from P2. Along with the compatibility equation, q_11 and q_21 are obtained. Back to superposition, the shear flows of P1 and P2 are added together to get the true shear flows. For part 2, the code uses the same moments as part 1 because nothing has changed. However, the change is in the moment equation. The moment equation now has three unknowns (q_12 and q_22 and q_32) so it alone is not sufficient in solving for the P1 shear flows. This is where the compatibility equation comes in. We know that the twist angle of the first cell is equal to the twist angle of the second and third cells. The moment and compatibility equations are arranged to that it has coefficients multiplied to q_12 and q_22 and q_32. Then, the equations were arranged in matrix form and solved for in Matlab. Then, superposition was used to find the true shear flow in the three-cell case.