Tuesday, 29 November 2016

homework and exercises - How to compute the inertia tensor ${bf{J}} _{Omega}$ of a body of revolution


Suppose that $\Omega$ is a body of revolution of the function $y=f(x), a\le x \le b$ around the $x$-axis, where $f(x)>0$ is continuous.



How to compute the inertia tensor ${\bf{J}} _{\Omega}$?


After computing ${\bf{J}} _{\Omega}$, how to solve ${\bf{J}} _{\Omega} \dot w={\bf{J}} _{\Omega}w \times w$?



Answer



I will answer the mass moment of inertia tension question. For an infinitesimal clump of mass ${\rm d}m$ located at $\vec{r}$ its effect on the inertia tensor is $${\rm d}{\bf J} = -[\vec{r}\times][\vec{r}\times]{\rm d}m$$


where $[\vec{r}\times]$ is the skew symmetric cross product operator $$ \begin{pmatrix}x\\y\\z \end{pmatrix}\times = \begin{bmatrix}0&-z&y\\z&0&-x\\-y&x&0\end{bmatrix}$$


For a axisymmetric part $\vec{r}(x,y,\theta) = (x,y \cos\theta,y \sin \theta)$ where $x=a\ldots b$, $y=0\ldots f(x)$ and $\theta=0\ldots 2\pi$.


The total mass is $m = \iiint \rho {\rm d}V$ where ${\rm d}V = (y{\rm d}\theta)({\rm d}y)({\rm d}x)$ $$m = \int \limits_a^b \int \limits_0^{f(x)} \int \limits_0^{2\pi} \rho\,y{\rm d}\theta\,{\rm d}y\,{\rm d}x$$ $$\rho = \frac{m}{\pi \int_a^b {f(x)}^2\,{\rm d}x}$$


The inertia tensor is ${\bf J}=\rho \iiint -[\vec{r}\times][\vec{r}\times]\,{\rm d}V$. You can do some math to find that the inner integral is


$$\int \limits_0^{2\pi} -[\vec{r}\times][\vec{r}\times]\,{\rm d}\theta = \begin{bmatrix} 2\pi y^2&0&0\\0&\pi(2 x^2+y^2)&0\\0&0&\pi(2 x^2+y^2)\end{bmatrix} $$


In the end, I get $${\bf J} = \rho\, \begin{bmatrix} \frac{\pi}{2}\int_a^b{f(x)}^4\,{\rm d}x&0&0\\ 0&\frac{\pi}{4}\int_a^b\left({f(x)}^4+4 x^2 {f(x)}^2\right)\,{\rm d}x &0\\0&0&\frac{\pi}{4}\int_a^b\left({f(x)}^4+4 x^2 {f(x)}^2\right)\,{\rm d}x \end{bmatrix}$$



No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...