Suppose I have a 3D rigid object on which some external forces act at various points located on it. The resulting motion would, in general, be the translation of center of mass plus rotation about the axis passing through Center of Mass. Velocity of center of mass can be easily found from summing external forces, so I have information about 'translational part'. I need to find the direction of Angular velocity or the direction of axis of rotation, assuming I know only about the location on the body where external forces act.
Suppose I find the torque about the center of mass (I chose COM because it simplifies the equations, no need to add pseudo torques).
But the direction of torque doesn't always point towards the axis of rotation (if it does then a nice proof using vectors would be a great help because I haven't been able to prove it), only the component of torque along the axis is proportional to magnitude of Angular acceleration, but that doesn't help because I want to know the direction of axis itself.
Finding angular momentum won't help either because angular momentum vector doesn't always point in the direction of angular velocity.
So if I know about the forces acting on the rigid body then how to predict the direction of rotational axis?
Basically knowing the direction of rotational axis passing through COM is important as once known I can simply take the component of net torque (about COM) along the axis to find angular acceleration and to fully analyze the motion.