72 Y. Shindo et al. Fig. 7.1 Rotor-bearing finite element model of single shaft turbopump of the bearings are including both Kand Cmatrices. In the right hand of the equation, fRD(ω) indicates the RD fluid forces at the operating speedωrpm. In the proposed model, RD fluid forces acting on the inducer, centrifugal impeller, and seal are modeled by using nondimensional normalized direct and cross-coupled RD coefficients and introduced as matrices into the node corresponding to the locations acting on the RD fluid forces as the following form: fRDi (ω) =− #MRDi mRDi mRDi MRDi $%¨qxi ¨qyi &−#CRDi cRDi cRDi CRDi $%˙qxi ˙qyi &−#KRDi kRDi kRDi KRDi $%qxi qyi & (7.2) Thomas force is adopted as the RD fluid force acting on the turbine [2]. In the numerical simulation of this study, RD properties of a single shaft turbopump rotor system was analyzed by using the proposed modeling. The configuration of the turbopump was adopted from reference [3]; however only an axial cross-sectional view was available. Therefore, in this study, each component size of both fuel turbopump and oxidizer pump is evaluated based on the performance analysis of the fuel pump. Figure 7.1 shows the rotor-bearing finite element model of the single shaft TP. 7.3 Results Lower three damped natural circular frequencies and their vibration mode shapes were successfully obtained by the proposed modeling. Due to the effect of both RD fluid and gyroscopic forces, each vibration mode was separated into backward and forward modes. In the Campbell diagram, mode crossing/veering phenomena appeared from 0 rpm to the normal operating speed 46,000 rpm. Lower three forward natural circular frequencies at 46,000 rpm are 1702 Hz, 882 Hz, and 1052 Hz, respectively. First forward natural circular frequency is quite higher than the second and third forward natural circular frequencies due to both RD fluid force and gyroscopic effect. Figures 7.2, 7.3, and 7.4 show these three forward mode shapes.
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