| Summary: | Nonlinear vibration control of the twelve-poles electro-magnetic suspension system was tackled in this study, using a novel control strategy. The introduced control algorithm was a combination of three controllers: the proportional-derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>P</mi><mi>D</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> controller, the integral resonant controller <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>I</mi><mi>R</mi><mi>C</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, and the positive position feedback <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo stretchy="false">(</mo><mrow><mi>P</mi><mi>P</mi><mi>F</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> controller. According to the presented control algorithm, the mathematical model of the controlled twelve-poles rotor was established as a nonlinear four-degree-of-freedom dynamical system coupled to two first-order filters. Then, the derived nonlinear dynamical system was analyzed using perturbation analysis to extract the averaging equations of motion. Based on the extracted averaging equations of motion, the efficiency of different control strategies (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mrow><mi>P</mi><mi>D</mi><mo>,</mo><mo> </mo><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi><mo>,</mo><mo> </mo><mi>P</mi><mi>D</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi><mo>,</mo><mo> </mo><mi>and</mi><mo> </mo><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi></mrow></mrow></mrow></semantics></math></inline-formula>) for mitigating the rotor’s undesired vibrations and improving its catastrophic bifurcation was investigated. The acquired analytical results demonstrated that both the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi></mrow></semantics></math></inline-formula> controllers can force the rotor to respond as a linear system; however, the controlled system may exhibit the maximum oscillation amplitude at the perfect resonance condition. In addition, the obtained results demonstrated that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula> controller can eliminate the rotor nonlinear oscillation at the perfect resonance, but the system may suffer from high oscillation amplitudes when the resonance condition is lost. Moreover, we report that the combined control algorithm (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>) has all the advantages of the individual control algorithms (i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi><mo>,</mo><mo> </mo><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi><mo>,</mo><mo> </mo><mi>P</mi><mi>D</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula>), while avoiding their drawbacks. Finally, the numerical simulations showed that the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>P</mi><mi>D</mi><mo>+</mo><mi>I</mi><mi>R</mi><mi>C</mi><mo>+</mo><mi>P</mi><mi>P</mi><mi>F</mi></mrow></semantics></math></inline-formula> controller can eliminate the twelve-poles system vibrations regardless of both the excitation force magnitude and the resonant conditions at a short transient time.
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