I have no engineering background at all, but when I bumped my small electric heater and it started whining and creating a burning smell, (since it's labor day with no stores open to buy a new one) I unplugged it and took it apart. Once I got down to the electric motor housing, I could spin the fan with a finger and hear the same whine, and noticed the axle (or whatever you call a rod that rotates a fan) attached to the fan was slightly crooked where it was housed within this piece.
Further inspection revealed that the brass-colored bearing (if that's the correct term) isn't securely seated within the star-shaped housing, but rather just held down by the thin tin flaps, and beyond that pressure moves freely, allowing it to seemingly cause the friction / disturbance creating the whine.
Is there a reason for the bearing to be held in place by this flimsy star chamber rather than a solid seating?
I tried to straighten it out and it appeared straight, but once I put it back together, the whine had only been reduced marginally, and the burning smell still occurred.