Geometric and Inertial Properties of a Hemispherical Shell



In [2]:

    
from miscpy.utils.sympyhelpers import *
init_printing()
m,r,R,R1,R2,th,ph = symbols('m,r,R,R_1,R_2,theta,phi')

The shell has interior radius $R_1$ and exterior radius $R_2$. The origin of the coordinate system is located at the center of the base (there is no cap on the shell - $O$ is in empty space). A differential mass element in the shell is located by spherical coordinates $r,\theta,\phi$ where $\theta$ is the aximuth and $\phi$ is the zenith angle.

$\mathbf{r}_{\mathrm{d}m/O}$ in Spherical Coordinates



In [3]:

    
rdmO = r*Matrix([cos(th)*sin(ph),sin(th)*sin(ph), cos(ph)]);rdmO









    Out[3]:





$$\left[\begin{matrix}r \sin{\left (\phi \right )} \cos{\left (\theta \right )}\\r \sin{\left (\phi \right )} \sin{\left (\theta \right )}\\r \cos{\left (\phi \right )}\end{matrix}\right]$$

Volume of Body



In [4]:

    
V = integrate(integrate(integrate(r**2*sin(ph),(r,R1,R2)),(th,0,2*pi)),(ph,0,pi/2)); V









    Out[4]:





$$- 2 \pi \left(\frac{R_{1}^{3}}{3} - \frac{R_{2}^{3}}{3}\right)$$

Center of Mass of Body



In [7]:

    
rGO = simplify(integrate(integrate(integrate(rdmO*r**2*sin(ph),(r,R1,R2)),(th,0,2*pi)),(ph,0,pi/2))/V);
mat2vec(rGO,'b')









    Out[7]:





$$\frac{3 \left(R_{1}^{4} - R_{2}^{4}\right) \mathbf{b}_3}{8 \left(R_{1}^{3} - R_{2}^{3}\right)}$$

COM of solid hemisphere



In [8]:

    
mat2vec(rGO.subs(R1,0),'b')









    Out[8]:





$$\frac{3 R_{2} \mathbf{b}_3}{8}$$



In [ ]:

Argument of Moment of Inertia Integral



In [12]:

    
tmp = simplify((rdmO.T*rdmO)[0]*eye(3) - rdmO*rdmO.T); tmp









    Out[12]:





$$\left[\begin{matrix}r^{2} \left(\sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} + \cos^{2}{\left (\phi \right )}\right) & - r^{2} \sin^{2}{\left (\phi \right )} \sin{\left (\theta \right )} \cos{\left (\theta \right )} & - \frac{r^{2} \left(\sin{\left (2 \phi - \theta \right )} + \sin{\left (2 \phi + \theta \right )}\right)}{4}\\- r^{2} \sin^{2}{\left (\phi \right )} \sin{\left (\theta \right )} \cos{\left (\theta \right )} & r^{2} \left(- \sin^{2}{\left (\phi \right )} \sin^{2}{\left (\theta \right )} + 1\right) & - \frac{r^{2} \left(\cos{\left (2 \phi - \theta \right )} - \cos{\left (2 \phi + \theta \right )}\right)}{4}\\- \frac{r^{2} \left(\sin{\left (2 \phi - \theta \right )} + \sin{\left (2 \phi + \theta \right )}\right)}{4} & - \frac{r^{2} \left(\cos{\left (2 \phi - \theta \right )} - \cos{\left (2 \phi + \theta \right )}\right)}{4} & r^{2} \sin^{2}{\left (\phi \right )}\end{matrix}\right]$$

Moment of Inertia of Body about $O$



In [13]:

    
Io = m/V*integrate(integrate(integrate(tmp*r**2*sin(ph),(r,R1,R2)),(th,0,2*pi)),(ph,0,pi/2)); Io









    Out[13]:





$$\left[\begin{matrix}- \frac{m \left(- \frac{4 \pi R_{1}^{5}}{15} + \frac{4 \pi R_{2}^{5}}{15}\right)}{2 \pi \left(\frac{R_{1}^{3}}{3} - \frac{R_{2}^{3}}{3}\right)} & 0 & 0\\0 & - \frac{m \left(- \frac{4 \pi R_{1}^{5}}{15} + \frac{4 \pi R_{2}^{5}}{15}\right)}{2 \pi \left(\frac{R_{1}^{3}}{3} - \frac{R_{2}^{3}}{3}\right)} & 0\\0 & 0 & \frac{m \left(\frac{2 R_{1}^{5}}{15} - \frac{2 R_{2}^{5}}{15}\right)}{\frac{R_{1}^{3}}{3} - \frac{R_{2}^{3}}{3}}\end{matrix}\right]$$

Parallel Axis Theorem to calculated MOI about $G$



In [14]:

    
Ig = simplify(Io  - m*((rGO.T*rGO)[0]*eye(3)-rGO*rGO.T)); Ig









    Out[14]:





$$\left[\begin{matrix}\frac{m \left(128 \left(R_{1}^{3} - R_{2}^{3}\right) \left(R_{1}^{5} - R_{2}^{5}\right) - 45 \left(R_{1}^{4} - R_{2}^{4}\right)^{2}\right)}{320 \left(R_{1}^{3} - R_{2}^{3}\right)^{2}} & 0 & 0\\0 & \frac{m \left(128 \left(R_{1}^{3} - R_{2}^{3}\right) \left(R_{1}^{5} - R_{2}^{5}\right) - 45 \left(R_{1}^{4} - R_{2}^{4}\right)^{2}\right)}{320 \left(R_{1}^{3} - R_{2}^{3}\right)^{2}} & 0\\0 & 0 & \frac{2 m \left(R_{1}^{5} - R_{2}^{5}\right)}{5 \left(R_{1}^{3} - R_{2}^{3}\right)}\end{matrix}\right]$$

MOI for solid hemisphere:



In [15]:

    
Ig.subs(R1,0)









    Out[15]:





$$\left[\begin{matrix}\frac{83 R_{2}^{2} m}{320} & 0 & 0\\0 & \frac{83 R_{2}^{2} m}{320} & 0\\0 & 0 & \frac{2 R_{2}^{2} m}{5}\end{matrix}\right]$$



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