How to calculate inertia tensor of composite shape?

Question

How can I calculate inertia tensor of composite shape. I have done math of moment of inertia, but not inertia tensor.

Can anyone help me with the calculation of this shape :

Eli · Accepted Answer · 2019-09-29T20:24:05.393

3

you can use the parallel axes theory :

$$I=I_1+I_2+I_3-m_1 \,\tilde{r}_{01}\,\tilde{r}_{01}-m_3\,\,\tilde{r}_{03}\,\tilde{r}_{03}$$

where :

$$\tilde{r}_{01}=\begin{bmatrix} 0 & -z & 0 \\ z & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$$

and $$\tilde{r}_{03}=\begin{bmatrix} 0 & +z & 0 \\ -z & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}$$

edited Sep 29 '19 at 20:24

answered Sep 28 '19 at 19:29

Eli

what are the value of $I_x$ and $I_y$ ? – Maifee Ul Asad Sep 29 '19 at 11:38
1

Sorry for that The result is I which is the inertia tensor $3\times3$ symmetric matrix – Eli Sep 29 '19 at 14:00
1

I took the * out , the result is the same as the first version. – Eli Sep 29 '19 at 20:25
Can you take a look here, please .. – Maifee Ul Asad Sep 30 '19 at 17:06
if I translate and move the whole object downward and then calculate the inertia tensor, what will be changed. I want to put the plane on the center of origin. I'm confused with those +/- signs. @Eli – Maifee Ul Asad Oct 24 '19 at 10:28
the result is the same if you take +z or -z because you multiply $\tilde{r},\tilde{r}$ which is the same for +z or -z – Eli Oct 24 '19 at 13:14

1 Answers1