This note provides an analytical proof of the optimal tuning of centrifugal pendulum vibration absorbers (CPVAs) to reduce in-plane translational and rotational vibration for a rotor with N cyclically symmetric substructures attached to it. The reaction forces that the substructures (helicopter or wind turbine blades, for example) exert on the rotor are first analyzed. The linearized equations of motion for the vibration are then solved by a gyroscopic system modal analysis procedure. The solutions show that the rotor translational vibration at order j is reduced when one group of CPVAs is tuned to order jN − 1 and the other is tuned to order jN + 1. Derivation of this result is not available in the literature. The current derivation also yields the better known result that tuning CPVAs to order jN reduces rotational rotor vibration at order j.

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