1540s, probably from Middle Dutch tromme "drum," common Germanic (cf. German Trommel, Danish tromme, Swedish trumma), probably of imitative origin. Not common before 1570s. Slightly older, and more common at first, was drumslade, apparently from Dutch or Low German trommelslag. Machinery sense attested from 1740, from similarity of shape.
1570s, from drum (n.). To drum (up) business, etc., is American English 1839, from the old way of drawing a crowd.
To broadcast emphatically and constantly; insistently feature: He's beating the drum for that pet idea (1600s+)