No, the precession angular velocity of a gyroscope cannot be equal to its self-rotation velocity. These two velocities refer to different types of motion exhibited by a gyroscope.
The self-rotation velocity, also known as the spin angular velocity, refers to the rotation of the gyroscope about its own axis. It determines how fast the gyroscope is spinning. The self-rotation velocity is typically generated by an external force or motor and is independent of the precession motion.
On the other hand, the precession angular velocity relates to the motion of the axis of the gyroscope as it precesses or changes direction. When a gyroscope experiences a torque, such as due to an external force applied at an angle to its spin axis, it undergoes precession. The precession angular velocity determines how fast the gyroscope's axis rotates or precesses around a different axis.
In a stable gyroscope, the precession angular velocity is generally much smaller than the self-rotation velocity. The ratio between the precession angular velocity and the self-rotation velocity is influenced by factors such as the geometry of the gyroscope and the applied external torques. However, it is not common for these two velocities to be equal in typical gyroscope systems.