3.12 - Slender body Approximation.

Estimating $m_{ij}$ of a slender 3D body using 2D strip-wise $M_{ij}$.

Idea `` $\underset{3D}{m_{ij}}$ = sum [ $\underset{2D}{M_{ij}(x)}$ contributions]''
e.g.

$$m_{33} = \int\limits_L {M_{33} \left( x \right)dx;} m_{22}, m_{33}, \mbox{\ etc.}$$

Yaw moment due to sway acceleration:

$$\mathop {m_{53} } = \int\limits_L {\left( { - x} \right)M_{33} \left( x \right)dx}$$

In general: Moment$_{5}$ = $(-x)$ force$_{3}(x)$ and $\dot {U}_3 \left( x \right) = \left( { - x} \right)\dot {\Omega }_5$

$$\mathop {m_{55} }\limits_ = \int\limits_L {\left( { - x} \ri... ... x \right)dx} = \int\limits_L {x^2M_{33} \left( x \right)dx}$$

Similarly for m$_{22}$, m$_{44}$, m$_{42}$, ...How about m$_{23}$, m$_{25}$ ?? (Work out the detail!!)
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