On early Sikorsky helicopters (S-55 series) they had problems of vertical oscillations but it was not pilot induced. It was determined that if the pilot did not have sufficient friction on the collective the weight of the flight control system would tune into the frequency of the helicopter movement moving up and down causing the servo valves to displace either in or out depending on the direction of movement of the control linkage. It only took 1-2 thousandths of an inch displacement to get the servo to move. They solved the problem by placing short but very strong springs across the sloppy link and the combined strength of the two springs on the servo(s) was sufficient to offset the mass of the flight control system so the pilot could fly hands off on the collective with minimal friction. A similar problem existed on the S-58 series and this problem was solved by the inclusion of a small force gradient spring in the collective linkage. The spring could be adjusted to support the weight of the collective and the interconnecting linkage to the auxiliary servo pack. The aux servo was fairly close coupled to the upper linkage and the primary servos. If the spring were adjusted too much in one direction the collective would rise and if adjusted in the opposite direction the collective would fall.
On later Sikorsky designs (S-61) they relocated the (Aux) servo system relative to the primary system and I do not know if they maintained the collective adjustment spring.
On the S-55 series they had another control related problem. When the helicopter was flown with the boost off the vertical feedback forces on the lateral linkages was excessive which made for difficult precision control. They installed an aux servo that was powered from the engine oil system. Upon the loss of boost pressure the engine oil powered servo was activated and it dampened out the lateral feedback.