• Optics and Precision Engineering
  • Vol. 32, Issue 4, 524 (2024)
Junfeng LI, Xiangbo XU*, Shao CHEN, Xianzhang WANG..., Lei FU and Yahui ZHU|Show fewer author(s)
Author Affiliations
  • School of Technology, Beijing Forestry University, Beijing100083, China
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    DOI: 10.37188/OPE.20243204.0524 Cite this Article
    Junfeng LI, Xiangbo XU, Shao CHEN, Xianzhang WANG, Lei FU, Yahui ZHU. Harmonic vibration suppression in active magnetic bearing systems with adaptive notch filter and repetitive control[J]. Optics and Precision Engineering, 2024, 32(4): 524 Copy Citation Text show less

    Abstract

    Harmonic vibrations caused by rotor mass imbalance and sensor runout, the main disturbances in magnetically suspended rotor systems. To suppress these disturbances, a compound control method based on repetitive control and variable phase adaptive notch filter feedback was proposed. Firstly, by establishing a model of the magnetic suspended rotor system, the generation mechanism of different disturbing vibration forces was analyzed. Then, taking the X-direction as an example for analysis, an inserted repetitive controller was designed to suppress the harmonic vibration caused by sensor runout. An adaptive notch filter was used to extract the synchronous signal online to adaptively compensate imbalance. The stability of the system was maintained by varying the phase angle at different frequencies, and to compensate the same-frequency displacement stiffness, so that the system can effectively suppress the harmonic vibration. Finally, the proposed control method was verified by simulation and experiment. The experimental results show that the first, third and fifth harmonic vibrations are reduced by 94.4%, 90.4% and 85.9%, respectively. The harmonic vibrations can be effectively suppressed using the proposed composite control method, whose effectiveness well verified.
    mx..=fx+fi(1)

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    fx=kxx+kiix(2)

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    fi=meΩ2cos(Ωt+ϕ)(3)

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    X(t)=x(t)-fd(t)(4)

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    xs(t)=x(t)+xsr(t)(5)

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    xsr(t)=i=1ncisin(iΩt+θi)(6)

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    ix=-kskadGc(s)Gw(s)X(s)+fd(s)+xsr(s)(7)

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    fx=ms2X(s)=kxX(s)+fd(s)+kiix(s)=kxX(s)+fd(s)-kikskadGc(s)Gw(s)X(s)+fd(s)+xsr(s).(8)

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    Gf(s)=G1(s)G2(s)(9)

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    Grc(s)=KrcGf(s)Q(s)e-T1s1-e-Ts(10)

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    ixD(s)=-KasGc(s)Gw(s)1-Grc(s)Gw(s)+Gc(s)Gw(s)Gp(s)Kas.(11)

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    ixD(s)=A1A2-A3(12)

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    A1=-KasGc(s)Gw(s)(1-e-Ts)(13)

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    A2=1+Gc(s)Gw(s)Gp(s)Kas(14)

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    A3=1+KrcGw(s)Gf(s)Q(s)eT0s+Gc(s)Gw(s)Gp(s)Kase-Ts.(15)

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    Gw'(s)=Gw(s)1-Gw(s)Grc(s)(13)

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    GANF(s)=εsinθs2+2Ωεcosθs+ε2Ω-Ω2εsinθ(Ω-εsinθ)(s2+Ω2)(14)

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    x1=εsinθs2+2Ωεcosθs+ε2Ω-Ω2εsinθΩs2+2εcosθs+Ω2-2Ωεsinθ+ε2u1(15)

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    ixD(s)=(s2+Ω2)(Ω-εsinθ)B1B2+B3(16)

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    B1=1+KasGp(s)Gc(s)Gw'(s)(20)

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    B2=εsinθs2+2Ωεcosθs+ε2Ω-Ω2εsinθ(21)

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    B3=-Kas(s2+Ω2)(Ω-εsinθ)Gp(s)Gc(s)Gw'(s)(22)

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    rx=kadksxsr(s)+x(s)(17)

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    x(s)s=jΩ=fd(jΩ)x(s)s=jkΩ=0,k=2,3,4,(18)

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    xsr(s)s=jΩ=xsr(jΩ)xsr(s)s=jkΩ=xsr(jkΩ),k=2,3,4,(19)

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    rx(s)s=jΩ=Kasxsr(jΩ)+fd(jΩ)rx(s)s=jkΩ=Kasxsr(jkΩ),k=2,3,4,(20)

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    cfxf=A2ss2+Ω2(21)

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    x12+x22=A2(22)

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    c=ε1A2ss2+Ω2ix-kxKaskix11x12+x22-ixGrc(s)(23)

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    ix=Gw(s)u11+GANF(s)Gc(s)+c(24)

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    E(s)=ixu1=C1Gc(s)Gw(s)1+GANF(s)-C2(25)

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    C1=s2+Ω2s2+Ω2-ε1s-s2+Ω2Grc(s)Gw(s)(32)

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    C2=ε1ss2+Ω2kxKaskiGANF(s)Grc(s)Gw(s)1+GANF(s)(33)

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    ix(s)s=jΩ=-kxkiKasrx(jΩ)=-kxkifd(jΩ)ix(s)s=jkΩ=0,k=2,3,4,(26)

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    fx(s)s=jΩ=kxfd(jΩ)+kiix(jΩ)=0fx(s)s=jkΩ=0,k=2,3,4,(27)

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    S0(s)=Gw(s)1+Gc(s)Gw(s)Gp(s)Kas(28)

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    S0(s)s=jω=L(ω)ejθa(ω)(29)

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    Gf(s)s=jω=Kb(ω)ejθbω(30)

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    π2<γ<3π2(31)

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    0<Krc<2min|cosγ|max{Kb(ω)L(ω)}(32)

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    B2+B3=0(33)

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    (Ω-εsinθ)(s2+Ω2)+B4S1(s)=0(34)

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    B4=εsinθs2+2Ωεcosθs+ε2Ω-Ω2εsinθ(43)

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    S1(s)=11+KasGp(s)Gc(s)Gw'(s)(44)

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    sεε=0,s=jΩ=-(cosθ-jsinθ)S1(jΩ)(35)

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    -π2<argS1jΩ-θ<π2(36)

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    ixD(s)=-KasE(s)1+KasE(s)Gp(s)(37)

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    1+KasE(s)Gp(s)=0(38)

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    s2+Ω2-ε1S2(s)s=0(39)

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    S2(s)=Grc(s)Gw(s)D1+GANF(s)Grc(s)Gw(s)Gp(s)kx/ki1+GrcGw(s)D1+KasGc(s)Gw(s)Gp(s)(50)

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    D1=1+GANF(s)(51)

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    sε1ε1=0,s=jΩ=12S2(jΩ)(40)

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    π2<argS2jΩ<3π2(41)

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    2.8<N1<5.9(42)

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    Gf(s)=G1(s)G2(s)=s+1s0.002s+10.0001s+12(43)

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    Krc<0.428(44)

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    Junfeng LI, Xiangbo XU, Shao CHEN, Xianzhang WANG, Lei FU, Yahui ZHU. Harmonic vibration suppression in active magnetic bearing systems with adaptive notch filter and repetitive control[J]. Optics and Precision Engineering, 2024, 32(4): 524
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