Grinding noise from the drum during the wash and spin cycles usually suggest that the drum bearings have worn.
Should a washing machine drum moves back and forth. The tub can swing back and forth a lot when empty. But if we go over to this machine and do the same thing you can see theres virtually no movement between the edge of the drum and the door seal. The main outer drum is on suspension and springs so it should move back and forth and side to side but the inner drum shouldnt move front to back even a few millimetres because the drum pulley is fixed at the end.
The tub moves back and forth agitating the clothes and supplying the mechanical action that scrubs out stains. To check turn the drum back and forth. I personally think that theres too much back and forth motion on the drum rather than side to side or indeed up and down.
If I go to this machine and press against the top is the drum you can see how much the drum moves away from the door seal here. Hot and cold water enter through the detergent tray at the top. Agitators might be a better choice for industrial or commercial use hotels salons but for home use it is better to stick with an impeller as you might have delicate clothing to wash.
The answer is both yes and noIt actually depends on the kind of washing machine in questionThe problem is the motionA washing machine need some kind of an engine or motor in order to move. Grinding noise from the drum during the wash and spin cycles usually suggest that the drum bearings have worn. If your washing machine does wobble from side to side by merely repositioning and tightening the levelling legs should be.
Incidentally if your machine is free standing it is important the adjustable feet are adjusted correctly and locked into position. To check turn the drum back and forth. Thats because when the clothes spin they can make the drum shake violently.
Once it accelerates that wobble with disappear totally. Should a washing machine drum moves back and forth. First check that the washer is level.