| This work investigates the effects of gravity on the dynamic response of centrifugal pendulum vibration absorbers (CPVAs). This analytical study considers small amplitude motions of the absorbers so that linear vibration tools can be applied. The motivation of the study is to determine the behavior of CPVAs at low rotor speeds, where gravity effects can be comparable to those of rotation. The main goal of the present study is to predict patterns that were observed in the response of systems with several symmetrically placed absorbers, and to use more sophisticated analysis tools for symmetric systems, namely circulant matrices, to investigate the linearized version of the model. A mathematical model is developed using Lagrange's equations for a disk rotating about a fixed horizontal axis and N point masses cyclically arranged on the rotor that can move along paths relative to the rotor. Gravity provides both direct and parametric excitation to these pendulum masses at order one, whereas the torque applied to the rotor is at order n. The equations are linearized and non-dimensionalized for analysis. The number of distinct groups of absorbers with identical but phase-shifted waveforms is considered, and it is shown that this grouping behavior depends on the engine order n and the ratio N/n. Models with and without the effects of parametric n excitation are considered, and it is shown that parametric excitation leads to resonant effects when n = 1 and n = 2. It is shown that the rotor is affected only by the order n component of the absorber responses, because of the symmetries of the response at order one from gravity. These results pro- vide useful information about absorber behavior and can be used to assess potential problems that may arise from gravitational effects. |
