# How do you combine two rigid bodies into one?

With respect to some fixed frame of reference, given the inertial tensors, positions, orientations, and angular and linear velocities of two rigid bodies, how do you combine them to make a single rigid body?

Positions (center of mass in global frame): $x_1$, $x_2$
Orientations (rotation from canonical orientation in global frame): $R_1$, $R_2$
Inertial tensors (in body's frame) $I_1$, $I_2$
Total masses (scalar): $m_1$, $m_2$
Linear velocities (global frame): $v_1$, $v_2$
Angular velocities around center of mass (global frame): $\omega_1$, $\omega_2$

The new position and mass are easy, of course:

$m_f = m_1 + m_2$
$x_f = \frac{x_1m_1 + x_2m_2}{m_f}$

The canonical orientation for the combined body isn't really defined; so we can just make it the identity matrix:

$R_f = \left| \begin{array}{ccc} 1 & 0 & 0\\ 0 & 1 & 0\\0 &0&1\end{array}\right|$

I can then combine the inertial tensors into the new local frame of reference:

$t_1 = x_f - x_1$ (translation of $I_1$)
$J_1 = \left| \begin{array}{ccc} -(t_{1y}^2+t_{1z}^2)&t_{1x}t_{1y} &t_{1x}t_{1z} \\ t_{1x}t_{1y} &-(t_{1x}^2+t_{1z}^2)&t_{1y}t_{1z} \\ t_{1x}t_{1z} &t_{1y}t_{1z} &-(t_{1x}^2+t_{1y}^2) \end{array} \right|$ (unscaled change)
And likewise for $J_2$

$I_f = (R_1I_1R_1^\intercal+ m_1J_1)+ (R_2I_2R_2^\intercal+m_2J_2)$

I think it's mostly right up to there. How do I find $\omega_f$ and $v_f$ so that all of the energy is accounted for?

Attempting to answer my own question:

Can I treat the two bodies as point masses and then combine their velocities according to conservation of momentum? It feels wrong.

$v_f = \frac{m_1v_1 + m_2v_2}{m_f}$

It feels even more wrong to find the joint angular momentum:

$L_f = I_1\omega_1+t_1\!\!\times\!\!(m_1v_1)+I_2\omega_2+t_2\!\!\times\!\!(m_2v_2) = I_f\omega_f$
$\omega_f = I_f^{-1}L_f$

Is that it?

• Possibly related: physics.stackexchange.com/q/19724/2451 Commented Jan 20, 2012 at 19:48
• For what purpose are you combining the bodies? Is this an inelastic collision?
– Rick
Commented May 23, 2016 at 13:42
• At least as far as rotations are concerned, I believe this problem is essentially solved in Repka, J., and N. Wildberger. "Invariant measure on sums of symmetric 3* 3 matrices with specified eigenvalues." Journal of Physics A: Mathematical and General 23.24 (1990): 5717, a copy of which is available from ResearchGate here: tinyurl.com/ydz944tu Commented Sep 23, 2018 at 21:03
• How can you just suddenly consider two bodies with independent velocities and rotation suddenly as a combined rigid body. That needs further specification of what’s going on. Was there some kind of collision and they stuck together? Commented Mar 6, 2021 at 7:31