Noise and vibration reduction is a critical concern in planetary gear applications. During the design process, system parameters are varied to evaluate alternative design choices, avoid resonances, optimize load distribution, and reduce weight. It is important to characterize the effects of parameter variations on the natural frequencies and vibration modes for effective vibration tuning. In planetary gear dynamic models (Figure 1), the key design parameters include the mesh stiffness, support/bearing stiffness, component mass, and moment of inertia. Some plots of natural frequencies versus planetary gear parameters are presented by Cunliffe et al. (1974), Botman (1976), Kahraman (1994), and Saada and Velex (1995). Lin and Parker (1999a) analytically characterized the unique, highly-structured properties of planetary gear natural frequency spectra and vibration modes. They also derived simple, closed-form expressions for the sensitivities of natural frequencies and vibration modes to these parameters (Lin and Parker, 1999b). The natural frequency plots in the above literature show natural frequency veering phenomena where two eigenvalue loci approach each other as a parameter is varied but then abruptly veer away like two similar charges repelling (Figure 2a). The phenomenon has been extensively studied (Leissa, 1974; Perkins and Mote, 1986; Pierre, 1988; Chen and Ginsberg, 1992; Happawana et al., 1998). The vibration modes of the veering eigenvalues are strongly coupled and undergo dramatic changes in the veering neighborhood. Eigenvalue veering is also related to mode localization that can occur when disorder is introduced into nominally symmetric systems like turbine blades, space antennae, multi-span beams and other structures (Pierre, 1988). In the case of especially sharp veering, it is sometimes difficult to distinguish between intersection and veering just by observing eigenvalue plots. Curve veering/crossing complexity obstructs the tracing of eigenvalue loci under parameter changes. Also, when multiple curves veer or intersect close together (Figure 3), strong modal coupling (and associated operating condition response changes) occurs that is not identifiable from frequency loci plots. The objective of this work is to analytically characterize the rules of eigenvalue veering in planetary gear vibration. The special veering patterns derived here help to trace natural frequencies and examine impacts of design parameters on vibration. These veering patterns, combined with the unique modal properties (Lin and Parker, 1999a) and the eigensensitivies analysis (Lin and Parker, 1999b), provide designers considerable insight into planetary gear free vibration.

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