The Cone component models a homogeneous conical rigid body along a given axis with a predefined density. Based on the properties, i.e., axial unit vector, length, radius, and density, the center of mass, total mass, and moments of inertia are calculated for this rigid body.

The two end frames of the cone have the same orientation. The translation vectors $L \mathrm{e\_\_axis}$ and $\mathrm{L\_\_com} \mathrm{e\_\_axis}$ w.r.t. frame_a define the frame_b and the center of mass frame, respectively.

Center of mass location is calculated as

$\mathrm{L\_\_com}=\frac{L\left({\mathrm{R\_\_a}}^2+2\mathrm{R\_\_a}\mathrm{R\_\_b}-{\mathrm{R\_\_ai}}^2-2\mathrm{R\_\_ai}\mathrm{R\_\_bi}+3{\mathrm{R\_\_b}}^2-3{\mathrm{R\_\_bi}}^2\right)}{4{\mathrm{R\_\_a}}^2+4\mathrm{R\_\_a}\mathrm{R\_\_b}-4{\mathrm{R\_\_ai}}^2-4\mathrm{R\_\_ai}\mathrm{R\_\_bi}+4{\mathrm{R\_\_b}}^2-4{\mathrm{R\_\_bi}}^2}$

Cone mass is calculated as

$m=\frac{\mathrm{\pi }L \mathrm{\ρ} \left({\mathrm{R\_\_a}}^2+\mathrm{R\_\_a}\mathrm{R\_\_b}-{\mathrm{R\_\_ai}}^2-\mathrm{R\_\_ai}\mathrm{R\_\_bi}+{\mathrm{R\_\_b}}^2-{\mathrm{R\_\_bi}}^2\right)}{3}$

where the cone material density, ρ, can be defined using the "Select density" parameter. This parameter lets the user either enter a value or select among predefined material densities.

Figure 1: Different options for the "Select density" property

Assuming the default direction of $\left[1\,0\,0\right]$ for the cylinder axis, the moments of inertia expressed from the center of mass frame are

$\mathrm{I\_\_xx}=\frac{\mathrm{\pi }\mathrm{\ρ} L \left({\mathrm{R\_\_a}}^4+{\mathrm{R\_\_a}}^3\mathrm{R\_\_b}+{\mathrm{R\_\_a}}^2{\mathrm{R\_\_b}}^2+\mathrm{R\_\_a}{\mathrm{R\_\_b}}^3-{\mathrm{R\_\_ai}}^4-{\mathrm{R\_\_ai}}^3\mathrm{R\_\_bi}-{\mathrm{R\_\_ai}}^2{\mathrm{R\_\_bi}}^2-\mathrm{R\_\_ai}{\mathrm{R\_\_bi}}^3+{\mathrm{R\_\_b}}^4-{\mathrm{R\_\_bi}}^4\right)}{10}$

$\mathrm{I\_\_yy}\=\mathrm{I\_\_zz}=\frac{3L\mathrm{\ρ}\mathrm{\pi }\left(4{\mathrm{R\_\_ai}}^6+8{\mathrm{R\_\_ai}}^5\mathrm{R\_\_bi}+\left(L^2-4{\mathrm{R\_\_a}}^2-4\mathrm{R\_\_a}\mathrm{R\_\_b}-4{\mathrm{R\_\_b}}^2+12{\mathrm{R\_\_bi}}^2\right){\mathrm{R\_\_ai}}^4+4\mathrm{R\_\_bi}\left(L^2-{\mathrm{R\_\_a}}^2-\mathrm{R\_\_a}\mathrm{R\_\_b}-{\mathrm{R\_\_b}}^2+3{\mathrm{R\_\_bi}}^2\right){\mathrm{R\_\_ai}}^3+\left(12{\mathrm{R\_\_bi}}^4+\left(10L^2-4{\mathrm{R\_\_a}}^2-4\mathrm{R\_\_a}\mathrm{R\_\_b}-4{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_bi}}^2-4{\mathrm{R\_\_a}}^4-4{\mathrm{R\_\_a}}^3\mathrm{R\_\_b}+\left(-2L^2-4{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_a}}^2+\left(-4L^2\mathrm{R\_\_b}-4{\mathrm{R\_\_b}}^3\right)\mathrm{R\_\_a}-4{\mathrm{R\_\_b}}^4-\frac{26L^2{\mathrm{R\_\_b}}^2}{3}\right){\mathrm{R\_\_ai}}^2-4\mathrm{R\_\_bi}\left(-2{\mathrm{R\_\_bi}}^4+\left(-L^2+{\mathrm{R\_\_a}}^2+\mathrm{R\_\_a}\mathrm{R\_\_b}+{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_bi}}^2+{\mathrm{R\_\_a}}^4+{\mathrm{R\_\_a}}^3\mathrm{R\_\_b}+\left(L^2+{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_a}}^2+\left(\frac{2}{3}{L}^2\mathrm{R\_\_b}+{\mathrm{R\_\_b}}^3\right)\mathrm{R\_\_a}+L^2{\mathrm{R\_\_b}}^2+{\mathrm{R\_\_b}}^4\right)\mathrm{R\_\_ai}+4{\mathrm{R\_\_bi}}^6+\left(L^2-4{\mathrm{R\_\_a}}^2-4\mathrm{R\_\_a}\mathrm{R\_\_b}-4{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_bi}}^4+\left(-4{\mathrm{R\_\_a}}^4-4{\mathrm{R\_\_a}}^3\mathrm{R\_\_b}+\left(-4{\mathrm{R\_\_b}}^2-\frac{26L^2}{3}\right){\mathrm{R\_\_a}}^2+\left(-4L^2\mathrm{R\_\_b}-4{\mathrm{R\_\_b}}^3\right)\mathrm{R\_\_a}-2L^2{\mathrm{R\_\_b}}^2-4{\mathrm{R\_\_b}}^4\right){\mathrm{R\_\_bi}}^2+4{\mathrm{R\_\_a}}^6+8{\mathrm{R\_\_a}}^5\mathrm{R\_\_b}+\left(L^2+12{\mathrm{R\_\_b}}^2\right){\mathrm{R\_\_a}}^4+\left(4L^2\mathrm{R\_\_b}+12{\mathrm{R\_\_b}}^3\right){\mathrm{R\_\_a}}^3+\left(10L^2{\mathrm{R\_\_b}}^2+12{\mathrm{R\_\_b}}^4\right){\mathrm{R\_\_a}}^2+\left(4L^2{\mathrm{R\_\_b}}^3+8{\mathrm{R\_\_b}}^5\right)\mathrm{R\_\_a}+L^2{\mathrm{R\_\_b}}^4+4{\mathrm{R\_\_b}}^6\right)}{240{\mathrm{R\_\_a}}^2+240\mathrm{R\_\_a}\mathrm{R\_\_b}-240{\mathrm{R\_\_ai}}^2-240\mathrm{R\_\_ai}\mathrm{R\_\_bi}+240{\mathrm{R\_\_b}}^2-240{\mathrm{R\_\_bi}}^2}$

The right-hand side of these equations will interchange if another axial unit vector is specified.