Coil Doesn't Chatter with AC supply

No matter which pole, a magnet still attracts a piece of iron.

Whenever I build a coil for a relay I solder a rectifier diode in the circuit

Wouldn't that make matters worse, by removing half the waveform so that half the time no magnetic field was induced?

<< but a good AC coil with a shading coil will >>

I am sure you mis-typed it [shaded-pole]

The term shaded pole refers to motor type,whose inductance self limits the current to the stator and prevents burn out, even if rotor is stalled.These are commonly found used as clock motors.Examine one, and you will see the relatively large copper shunt (shading pole) wrapped around the laminations.In a relay, the term shading coil refers to the single turn of copper in front of the coil, embedded into the core.

Interestingly,if you remove the copper shunt from a shaded pole motor, it will self destruct due to high rpm,so in this case, the shunt serves a different purpose.

<<< and you will see the relatively large copper shunt (shading pole) wrapped around the laminations.In a relay, the term shading coil refers to the single turn of copper in front of the coil, embedded into the core.>>>

A turn does not only apply to a thin wire wrapped around anything or in air.

A thick Strip wound is also a TURN even if it is a shorted, not a shunt. If it is fixed

around a portion of a lamination of motor / relay or contactor it is called a shaded Pole.... device /component.

Just see a very basic thing.

Every AC relay coil or solenoid has a shading coil in the armature, or stator.You can easily see this as a single turn of copper embedded in it.When the ac magnetic field collapses,an opposing field of opposite polarity is generated in the shading coil.This continues to hold the armature in place during the zero crossing interval.Get a small DC relay, and an AC relay of same size.Look closely at the coil and you will see a single turn of copper in the AC relay, but not the DC one.This is the shading coil.If it cracks,the relay will hum.

Because, they do have an accessory called "shading ring" in the cores with AC coils; it produces a supplementary flux, lagging the main flux by 90 degrees, which prevents the contactor from "dropping OFF" during the voltage zeroes.

Also, it is incorrect that the coil would "release" during the negative half cycle. In the negative half cycle too the core gets magnetised, but in the reverse direction.

If at all the coild should release, then it should be during the voltage zeroes in every half cycle. But, that too is prevented by the shading ring.

Coil always pull no matter the direction of the current inside coil wire if the plunger is made of a ferromagnetic material such as an iron. If the plunger is made off a permanent magnet which has a negative and positive pole then the direction of the current in the coil matters.

But why the coil starts chatter when only it is getting old ?

Clearances have changed or the shading coil has succumbed to repetitive stresses and has gone open or become loose.A particular motor starter relay I remember,it was a GE size 0,had a very heavy duty cycle of about once per second,24 hours a day,7 days a week.At about the 1 year interval, it would start to hum,and was replaced.

Examination of the armature revealed the armature laminations had been deformed and folded back into the shading coil, causing it to fracture.

Some of the really older relays had what were referred to as "slides", which were shims that could be replaced as they wore out.Of course, this was before we became a throw-away society.