User:Eas4200c.f08.wiki.c/homework report 3

=Meeting 14=

The analysis of a hollow circular cross section with thin walls (ie. t << a), is conducted as follows:

$$r_{i} = a$$ (inner radius)

$$r_{o} = b$$ (outer radius)

The equation for the polar moment is done by superimposing 'empty' circle on top of one solid circle, and the resulting equation is as follows:

$$J = \frac{1}{2}\pi (b^{4} - a^{4})$$

This can be expanded as:

$$J = \frac{1}{2}\pi (b - a)(b + a)(b^{2} + a^{2}$$

If an average radius of the cross section is then defined as:

$$\bar{r} = \frac{a + b}{2}$$

And it is noted that, since t is very small,

$$ b \cong \bar{r}$$

and

$$ a \cong \bar{r}$$

Therefore,

$$ a^{2} \cong \bar{r}^{2}$$

$$ a^{2} \cong \bar{r}^{2}$$

and noting that $$b - a = t$$

J can then be expressed as:

$$J = 2\pi t\bar{r}^{3}$$

J can then be expressed in terms of the average radius by noting that:

$$J = (2\pi ^{-\frac{1}{2}}t)(\pi \bar{r}^{2})^{3/2}$$

With $$\pi \bar{r}^{2} = \bar{A}$$ it's easily seen that J is proportional to $$\bar{A}^{3/2}$$, with $$(2\pi ^{-\frac{1}{2}}t)$$ being a proportionality factor.

=Meeting 15=

Matlab Project Discription
For HW 3, the team was asked to analyze a NACA 4-digit airfoil series. The discription of the problem can be seen here. A diagram showing the shape of the airfoil in this series can be found on this website. The group is asked to break up the airfoil into a number of discrete sections (ns) in order to analyze the geometry of the airfoil.

In order to complete this problem the student must first understand how to discretize an object into triangular segments in order to find the area of the object. A diagram of how this is done is provided below.



Using this, the following equations can be used in order to find the area of the shaded region.

$$d\vec{T} = \vec{r} \times d\vec{F} = q\vec{r} \times \vec{PQ}$$

Here, $$d\vec{F} = \vec{PQ}$$ and $$q\vec{r} \times \vec{PQ} = (2dA)\vec{i}$$

With

$$||\vec{PQ}|| = d\mathit{l} $$

Now that the basic foundation of the analysis has been discussed, the execution of the code will be done later in this report.

Torsion
Torsion of uniform, non circular bars leads to warping of the cross section, which is defined as an axial displacement along the x-axis (ie, along the bars length due to the conventions used in class) of a point on the deformed, or rotated, cross section. A schematic representation is shown below.



$$u_{y}$$ = the y-component of the displacement vector $$\vec{PP^{'}}$$

and

$$u_{z}$$ = the z-component of the displacement vector $$\vec{PP^{'}}$$

It should be noted (and easily seen) from the figure that $$\vec{PP^{'}}$$ is not on a circle centered at the origin with radius $$\vec{OP^{'}}$$ and $$\vec{OP^{'}}$$ in fact has a larger magnitude than $$\vec{OP}$$

This is due to the line $$\vec{PP^{'}}$$ having a vertical and horizontal displacement from $$\vec{PP^{'}}$$

However, since the angles associated with twist in torsion are likely to be relatively small, one can use small angle approximations and assume that the magnitude of $$\vec{OP} = \vec{OP^{'}} = R$$

$$u_{z} = Rsin\alpha \cong R\alpha$$

$$u_{y} = R(1 - cos\alpha )$$

With $$\alpha = 0$$, this makes $$u_{y} = 0$$ and $$u_{z} = \alpha R$$