Mechanical system (1) comprising a tappet (10) including a body (11), an insert (20) mounted in the body (11), a pin (30) extending along a first axis (X1), and a roller element (40) movable in rotation relative to the pin (30) around the first axis (X1). The insert (20) supports the pin (30), while the body (11) supports the insert (20), along at least one first direction (D1) parallel to a second axis (Y1) which is perpendicular to the first axis (X1). Said mechanical system is used in the cam drive of an injection pump or of a valve actuator.