In three 2010 papers, Tsujimoto et al. (2010, “Moment Whirl Due to Leakage Flow in the Back Shroud Clearance of a Rotor,” Int. J. Fluid Mach. Syst., 3(3), pp. 235–244), Song et al. (2010, “Rotordynamic Instability Caused by the Fluid Force Moments on the Backshroud of a Francis Turbine Runner,” Int. J. Fluid Mach. Syst., 3(1), pp. 76–79), and Song et al. (2010, “Rotordynamic Moment on the Backshroud of a Francis Turbine Runner Under Whirling Motion,” ASME J. Fluids Eng., 132, p. 071102) discussed and explained a novel destabilizing mechanism arising in both hydraulic turbines and the back surface of vertical pump impellers. The destabilizing mechanism can be explained via a reaction force-moment model that includes both the customary radial displacement vector of an impeller plus the pitch and yaw degrees of freedom. This coupling between radial displacements and tilt plus the coupling of the shaft support structure can create negative damping. In 1993, Verhoeven et al. (1993, “Rotor Instability of a Single Stage Centrifugal Pump, Supersynchronous Whirling at Almost Twice the Operating Speed, A Case History,” Proceedings of the 1st International Symposium on Pump Noise and Vibration, pp. 457–468) identified negative damping arising from U-shaped wearing-ring seals as causing a super-synchronous instability in a horizontal coke-crusher pump. However, several case studies have been presented of super-synchronously unstable pumps for which (until now) no explanation could be provided. Tsujimoto–Song started with a 2DOF model for a vertically suspended disk via a cantilevered shaft. They used an f = ma model for the lateral displacements of the disk and used flexibility coefficients to account for reaction forces and moments from the back shroud of the impeller. The present work starts with a 4DOF model that includes the disk's displacements and pitch and yaw degrees of freedom. The Guyan reduction is used to create two reduced 2DOF models: model A that retains the displacements and discards the rotations and model B that retains the rotations and discards the displacements. Model A produces a requirement for instability that is inconsistent with Tsujimoto–Song's experience and predictions. However, it is useful in predicting the reaction moments produced by a nominally planar precession of the impeller. The instability requirement of Model B is consistent with Tsujimoto's experience and predictions. A comparison of the predicted reaction moments of model A and Tsujimoto's reaction-moment data supports the instability predictions of model B (and Tsujimoto–Song) that the instability arises due to coupling between the displacement and rotation degrees of freedom in the 4 × 4 damping matrix.

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