If you apply a torque so that the system rotates about the A-B axis, you do change by a very slight amount the angular momentum about the A-B axis. That portion of angular momentum is for the most part ignored in this analysis. The more important effect is that you will change the direction of the angular momentum of the spinning gyroscope. The change in the angular momentum of the spinning gyro could very well point downward, i.e. ## \Delta \vec{L}= \vec{L}_2-\vec{L}_1 ##, from the spinning gyro, can readily occur in the minus z direction, if you push downward on the near side of the frame that holds the gyro. ## \\ ## Since you did not apply any torque in the z direction, total z angular momentum needs to be conserved, and there will be an increase in the angular momentum of the system in the +z-direction to offset this. Basically there will be an increase in ## \Omega ##. Otherwise you would need to apply a torque in the minus z-direction to prevent this increase in ## \Omega ##. ## \\ ## Additional note: In these gyroscope problems, you can often assume the gyro to be spinning quite rapidly, so that ## \vec{L}_1 ## and ## \vec{L}_2 ## are quite large and of equal amplitude. A small change in direction from ## \vec{L}_1 ## to ## \vec{L}_2 ## can result in a very large ## \Delta \vec{L} ##, whose direction can be readily computed.
