## 4 Simulation and Experimental Results
### Experimental Results
With the proposed algorithm, the value of the rotor displacement converges from 0.08 mm to 0.03 mm, a reduction of 62.50%, which is far less than the unilateral protective gap of the auxiliary bearing (0.2 mm). Moreover, in terms of the convergence time of different algorithms, the convergence time of the ANF-PSF algorithm is 3.67 s and the convergence time of the proposed algorithm is 2.85 s. The latter is 0.82 s faster than the former, a reduction of 22.34%. Figure 17: Rotor displacement with different algorithms. (**a**) Rotor displacement and the FFT with or without DTNFs during static suspension (\(\Omega=0\) rpm). (**b**) Rotor displacement with different algorithms at the rated speed (\(\Omega=\) 24,000 rpm). Note: the small image in the figure is an enlarged partial detail image. Figure 18: The time-domain waveform of End-B rotor displacement at 24,000 rpm. (**a**) End-B axis trajectory diagram with different algorithms. (**b**) Left view of the End-B axis trajectory diagram with different algorithms. At the rated speed of 24,000 rpm, the left view of the End-B axis trajectory diagram with different algorithms is shown in Figure 18b. Without any algorithm, with the ANF-PSF algorithm, and with the proposed algorithm, the maximum peak value of rotor displacement are 42.5%, 20%, and 7.5% of the double protective gap of the auxiliary bearing 0.4 mm, respectively. At rotational speeds of 12,000 rpm, 15,000 rpm, 18,000 rpm, and 21,000 rpm, FFTs of the rotor displacement in the BY channel with different algorithms are shown in Figure 19. With the ANF-PSF algorithm, the suppression of co-frequency synchronous vibration is approximately maintained between \(-21.57\) dB and \(-17.39\) dB, and the fluctuation range varies greatly with the rotor speed. With the proposed algorithm, the co-frequency synchronous suppression is further attenuated, being approximately maintained between \(-30.94\) dB and \(-30.56\) dB, and the fluctuation range varies relatively little with the rotor speed. Therefore, it is verified that the proposed algorithm can guarantee the consistency of vibration suppression at different rotational speeds. At the rated speed of 24,000 rpm, FFTs of the rotor displacement in the BY channel with different algorithms are shown in Figure 20. With the ANF-PSF algorithm, the amplitude of the synchronous frequency (SF) is reduced from \(-15.82\) dB to \(-17.36\) dB, and the amplitude of the nutation frequency (NF) is reduced from \(-58.25\) dB to \(-62.07\) dB. With the proposed algorithm, the amplitude of the SF is further reduced to \(-30.74\) dB, and the amplitude of the NF is further reduced to \(-71.65\) dB. The reason for this phenomenon is that after optimizing multiple notch parameters such as the bandwidth factor \(\varepsilon\), the depth factor \(\zeta\), and the width factor \(k\), the phase-lag angle introduced by the parallel co-frequency ANF and the cascaded multi-frequency improved DTNFs is limited, which leads to an increase in the phase stability margin of the entire AMB-rotor system. Structure improvement of notch filters and the multi-parameter optimization of the algorithm are not only limited to incomplete differential PID controllers but also can be combined with other advanced controllers, giving the approach strong engineering application and universality. Figure 16.0 The frequency-domain FFTs of rotor displacement at different rotational speeds in the BY channel with different algorithms. (**a**) 12,000 rpm. (**b**) 15,000 rpm. (**c**) 18,000 rpm. (**d**) 21,000 rpm.