Rotor-dynamic fluid force (RD fluid force) of turbomachinery is one of the causes of the shaft vibration problem. Bulk flow theory is the method for analyzing this RD fluid force, and it has been widely used in the design stage of machine. The conventional bulk flow theory has been carried out under the assumption of concentric circular shaft's orbit with a small amplitude. However, actual rotating machinery's operating condition often does not hold this assumption, for example, existence of static load on the machinery causes static eccentricity. In particular, when such a static eccentricity is significant, the nonlinearity of RD fluid force may increase and become non-negligible. Therefore, conventional bulk flow theory is not applicable for the analysis of the RD fluid force in such a situation. In this paper, the RD fluid force of the annular plain seal in the case of circular whirling orbit with static eccentricity is investigated. The case with both the significant static eccentricity and the moderate whirling amplitude is considered, and the perturbation analysis of the bulk-flow theory is extended to investigate the RD fluid force in such cases. In this analysis, the assumption of the perturbation solution is extended to both static terms and whirling terms up to the third order. Then, the additional terms are caused by the coupling of these terms through nonlinearity, and these three kinds of terms are considered in the extended perturbation analysis of the bulk flow theory. As a result, a set of nonlinear analytical equations of the extended perturbation analysis of the bulk flow theory, for the case with both the significant static eccentricity and the moderate whirling amplitude, is deduced. The RD fluid force for such cases is analyzed, and the occurrence of constant component, backward synchronous component, and super-harmonic components in the RD fluid force is observed in addition to the forward synchronous component. The representation of RD fluid force coefficients (RD coefficients) are modified for the case with significant static eccentricity, and the variation of RD fluid force coefficients for the magnitude of static eccentricity is analyzed. These analytical results of RD fluid force and its RD coefficients are compared with the numerical results using finite difference analysis and experimental results. As a result, the validity of the extended perturbation analysis of the bulk-flow theory for the case with both the significant static eccentricity and the moderate whirling amplitude is confirmed.

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