1998 - Draper Laboratory

Application to a MagneticBearing Gas Turbine EngineA finite-element model of a magnetic bearing rotor wasconstructed with 91 nodes and 90 elements. The majorcomponents of this rotor system are shown schematically inFigure 3, with the 12 actuator-sensor matrix locations.The forward (Table 1) and backward mode shapes have also beencomputed for the shaft critical speeds. The mode shapes of thefirst three whirl frequencies at the shaft first critical speed of2,551 rpm have been normalized to 1 and are shown in Figure5. The first three mode shapes at the second and third criticalspeed are similar and are not shown.Table 1. Forward whirl critical speeds.ThrustbearingFrontradialbearing &sensorStartergenerator4 compressordisksAftradialbearing& sensorTurbinedisk1 2 3 4 5 6 7 8 9 10 11 1213rd ModeFigure 3. Rotor components and sensor/actuator locations.UnitMagnitude02nd Mode1st ModeA rotordynamic analysis was performed using Eq. (12) and anisotropic radial bearing stiffness of 100,000 lb/in (17.51 x 10 6N/m) at each radial bearing. A starter-generator negativestiffness was applied to each end of the starter-generator,locations 4 and 5 in Figure 3, for a total equivalent stiffness of-70,000 lb/in (-12.26 x 10 6 N/m). The first four whirl frequencypairs have been plotted on a Campbell diagram in Figure 4.The upper line in each whirl speed pair indicates forward whirl,while the lower is backward whirl. The diagonal line representsequal frequency and rotor speed. The intersection of this linewith the whirl speed indicates a synchronous whirl condition.The occurrence of critical speeds at the various synchronouswhirl conditions is determined by a forced response analysis.Figure 5. Natural frequency mode shapes at 2,551 rpm.The actual magnetic bearing stiffness will vary with frequency.Therefore, critical speeds for various mechanical spring rates ofthe bearings were calculated. The resultant critical speed vsbearing stiffness is shown in Figure 6. No critical speeds exist inthe operating range of the rotor for bearing stiffness up to600,000 lb/in (105.1 x 10 6 N/m).600500-1ForeStarter-Generator Stiffness = -70,000 lb/in (-12.26x10 6 N/m)Aft500400Bearing Stiffness = 100,000 lb/in (17.51x10 6 N/m)Starter-Generator Stiffness = -70,000 lb/in (-12.26x10 6 N/m)Solid Line = Forward WhirlDashed Line = Backward WhirlFrequency (Hz)400300200100022,000 RPMOperating range14,400 RPM0 200,000 400,000 600,000 800,000 1,000,000Frequency (Hz)300200100006000Figure 4. Rotor components and sensor/actuator locations.12000Rotor Speed (RPM)18000Bearing Stiffness (lbf/in)Figure 6. Rotor critical speed map.The purpose of the development of the state-space matrices is touse them in simulations of the closed-loop response dynamics ofthe rotor. For this application, the mechanical bearing stiffnessand damping are zero in the state-space matrices or plant system.These stiffnesses will be provided by applying external forces asa function of the shaft displacement {q}. The results, withoutconstraints to ground, should then have four zero-frequencyrigid-body modes. In addition, the number of modes wasRotordynamic Modeling of an Actively Controlled Magnetic Bearing Gas Turbine Engine5