I recently repaired an older Whirlpool with a similar problem. There is a rubber clutch (for lack of a better term) that is positioned between the motor and the transmission. It is engaged by two three-pronged plastic pieces that seem to have been engineered to fail. They slip as they engage and the stress eventually causes them to fracture.
The part cost less than $20 and anyone who can use a screwdriver should be able to do the repair.
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