User:Eas4200c.f08.vqcrew.c/Homework 4/my

Torsional Analysis of Idealized Single-Cell Airfoil
It was determined in an earlier section that the following relationship is true for the idealized airfoil given in Figure ##:

$$\theta= \frac{111.29 q}{G}$$

The two relationships for the torque are also known:

$$T = GJ\theta\;$$

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

Combining the above two equations and solving for $$J\;$$ yields:

$$J = \frac{2q\bar{A}}{G \theta}$$

Substituting in the expression for $$\theta\;$$ results in the following expression:

$$J = \frac{2\bar{A}}{111.29}$$

The area $$\bar{A}\;$$ was previously calculated in HW3. Plugging in this value into the above yields the following value for $$J\;$$.

$$J = .1001\; m^{4}$$

Torsional Analysis of Two-Celled Rectangular Airfoil
Figure ## depicts a rectangular airfoil of two cells. It was found in an earlier section that the following three relationships are true:

$$T = T_1 + T_2 = 2\ (q_1\ \bar A_1 + q_2\ \bar A_2)$$

$$\theta_1 = \frac{1}{2\ G\ \bar A_1}\ (\frac{2\ q_1\ a}{t_1} + \frac{q_1\ c}{t_1} + \frac{(q_1 - q_2)\ c}{t_{12}})$$

$$\theta_2 = \frac{1}{2\ G\ \bar A_2}\ (\frac{2\ q_2\ a}{t_2} + \frac{q_2\ c}{t_2} + \frac{(q_2 - q_1)\ c}{t_{12}})$$

Plugging in the known variables available in the earlier section, and solve for $$q_{1}\;$$ and $$q_{2}\;$$ in terms of T, the following relationships can be determined:

$$q_{1} = 1.012 T\;$$

$$q_{2} = 1.577 T\;$$

Noting that, by definition, $$\theta = \theta_{1} = \theta_{2}\;$$, plugging the above into the equation for $$\theta_{1}$$ yields:

$$\theta = 1170.125 \frac{T}{G}$$

It is also known that $$J\;$$ in terms of $$\theta\;$$ can be expressed as:

$$J = \frac{T}{G\theta}$$

Which leades directly to solving for J:

$$J = \frac{1}{1170.125} = 8.55x10^{-4} m^{4}$$

MATLAB Airfoil Code Validation
Part of the first version of the MATLAB problem was to verify that the code was working properly by testing it both against a known area and also for replication of results. In the collapsible table below there is a "validation code" which calls part of the HW3 MATLAB code as a subfunction.

The resulting output from running the validation code is provided below.

The first section of the validation involved a "circular" airfoil. The first task was to determine the area of a circle of radius 1 meter at various numbers of contour segments. A table summarizing the above results is as follows:


 * {| border=1


 * bgcolor="red"|Number of Segments
 * bgcolor="#99ccff" style="text-align: center;" |20
 * bgcolor="#99ccff" style="text-align: center;" |40
 * bgcolor="#99ccff" style="text-align: center;" |60
 * bgcolor="#99ccff" style="text-align: center;" |80
 * bgcolor="#99ccff" style="text-align: center;" |100
 * bgcolor="red"|Calculated Area
 * bgcolor="#99ccff" |3.1045
 * bgcolor="#99ccff" |3.1285
 * bgcolor="#99ccff" |3.1344
 * bgcolor="#99ccff" |3.1369
 * bgcolor="#99ccff" |3.1383
 * }
 * }

The second task was to determine the $$ns_{max}\;$$, the number of segments at which the calculated area is within 1% of the actual area, which in this case is equal to 3.1416 square meters. This was done by calculating the area at various values of n and then comparing them. A plot of the results is given in Figure ##, where $$ns_{max}\;$$ and its corresponding area are denoted by a black square. For reference, the calculated value for $$ns_{max}\;$$ was found to be 23 segments, and the calculated area at this point was 3.1115 square meters, which differs from the actual value by .96%.

The final task for this section was to calculate the circle area at various locations of the origin point $$P_{0}\;$$. The three locations used were the coordinate system origin $$[0,0]\;$$, the circle center $$[1,0]\;$$, and an arbitrary external point $$[1,-4]\;$$. A table summarizing the results is shown below. As expected, the area is identical in all three cases.


 * {| border=1


 * bgcolor="red"|Coordinate of Origin Point
 * bgcolor="#99ccff" style="text-align: center;" |[0,0]
 * bgcolor="#99ccff" style="text-align: center;" |[1,0]
 * bgcolor="#99ccff" style="text-align: center;" |[1,-4]
 * bgcolor="red"|Calculated Area (square meters)
 * bgcolor="#99ccff" |3.1115
 * bgcolor="#99ccff" |3.1115
 * bgcolor="#99ccff" |3.1115
 * }
 * }

The second section of the validation involved a NACA 2415 airfoil of chord length c = 1 meter. The first task was to find $$ns_{max}\;$$ as with the circular airfoil. However, instead of comparing against a known area, the calculated area at a given number of segments is compared with the calculated area using one less segment, such that $$ns_{max}\;$$ is located when the difference between two successive area calculations is less than 1% of the current area calculation. A plot of the results is given in Figure ##. For reference, the calculated value for $$ns_{max}\;$$ was found to be 9 segments, and the calculated area at this point was 0.0244 square meters.

The second task was to show that the area calculation is valid at any choice of the origin point $$P_{0}\;$$. The three locations used were the coordinate system origin $$[0,0]\;$$, the airfoil centroid $$[.210,.008]\;$$ (previously calculated in HW3), and an arbitrary external point $$[.25,-1]\;$$. A table summarizing the results is shown below. Unlike with the circular case, the calculated areas are not identical for each case. However, a quick relative error calculation shows that the calculated values are all within 1% of each other, and such a small difference can be easily explained by the small numerical errors inherent in this kind of analysis.


 * {| border=1


 * bgcolor="red"|Coordinate of Origin Point
 * bgcolor="#99ccff" style="text-align: center;" |[0,0]
 * bgcolor="#99ccff" style="text-align: center;" |[.210,.008]
 * bgcolor="#99ccff" style="text-align: center;" |[.25,-1]
 * bgcolor="red"|Calculated Area (square meters)
 * bgcolor="#99ccff" style="text-align: center;" |.0244
 * bgcolor="#99ccff" style="text-align: center;" |.0246
 * bgcolor="#99ccff" style="text-align: center;" |.0247
 * }
 * }

Description of Airfoil Geometry and Equations


Figure ## shows the geometry of a generic NACA 4-digit airfoil. The various quantities are defined as follows:

$$LE\;$$ - The point corresponding to the leading edge of the airfoil. $$TE\;$$ - The point corresponding to the trailing edge of the airfoil. $$c\;$$ - The chord length, describing the length of the straight line connecting the leading edge to the trailing edge. $$m\;$$ - The maximum camber, corresponding to the first digit of the 4-digit series, and given in percentage of chord. $$p\;$$ - The location of maximum camber, corresponding to the second digit of the 4-digit series, and given in tenths of chord. $$t_{a}\;$$ - The maximum thickness, corresponding to the third and fourth digits of the 4-digit series, and given in percentage of chord. $$y_{u}\;$$ - The contour defining the airfoil's upper surface. $$y_{l}\;$$ - The contour defining the airfoil's lower surface. $$MCL\;$$ - The contour defining the airfoil's mean camber line. Referred to as $$z_{c}\;$$ in the airfoil equations.

A set of general equations used to generate the various contours for an arbitrary NACA 4-digit airfoil can be found here.

Torsional Constant Comparison Between Single- and Multi-celled Airfoils
The new facet to the MATLAB assignment deals with a comparison of single and multi-celled airfoils, in terms of the torsional constant $$J\;$$. The code for the single-celled airfoil is in the table below:



The first portion of the assignment was to determine $$ns_{max}\;$$, as in previous assignments. The result was a $$ns_{max}\;$$ value of 12 segments and a $$J\;$$ value of $$4.7543x10^{-6} m^{4}\;$$. The second portion involved generating a graph of torsional constant vs. number of segments. The result is shown in Figure ##.

The second portion of the assignment was to determine $$J\;$$ for a NACA2415 airfoil with 3 cells. The MATLAB code used to carry out this task is given in a table below.



The plotted airfoil, with partitions at quarter and three-quarter chord, is shown in Figure ##. The calculated $$J\;$$ for this case is $$8.6086x10^{-6} m^{4}\;$$.

Comparison:

The two calculated values for $$J\;$$ are very close to each other. The difference is likely a result of numerical errors during the generation of the torsional constant in the above MATLAB codes. Accounting for this effect, the overall result echoes the conclusion given in Chapter 3 of Sun: the torsional rigidity of a thin-walled, closed cross-section cannot be appreciably improved by compartmentalization