The shading coil is a rather large copper band around the laminations of the metal core of the contactor. When a magnetic field is present on the core from the main coil, it induces another current in the shading coil through basic transformer action. This current is about 90 degrees out of phase with the primary current and peaks when the when the current in the primary coil is crossing the zero point, therefore maintaing a magnetic field on the contactor keeping it closed duiring this time.
It the shading coil were to burn out, which happens quite often because of the large amount of current present on the coil, it cannot hold the contactor closed during the zero crossover causing a chatter to become obvious on the contactor.
From Wikipedia: If the coil is designed to be energized with AC, a small copper ring can be crimped to the end of the solenoid. This "shading ring" creates a small out-of-phase current, which increases the minimum pull on the armature during the AC cycle.[1]
