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Effect of bias current of active magnetic bearing on Sommerfeld effect characterization in an unbalanced rotor dynamic system

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Abstract

Sommerfeld effect is a nonlinear jump phenomenon which occurs in unbalanced rotor–motor system, and its characteristics provide an idea of power required to operate the system in the post-resonance region. In this article, the steady-state and full transient characterization of Sommerfeld effect in an unbalanced rotor–motor active magnetic bearing system is studied by considering the nonlinear bearing force of the active magnetic bearing (AMB) in the system equation. The homotopy perturbation method (HPM) is used to solve the nonlinear coupled second-order differential equations of the 16-pole leg AMB system analytically, and the steady-state vibration amplitude of the rotor dynamic system is calculated using the same method. The numerical power balance technique is used to study the steady-state Sommerfeld effect in the system, and its characterization provided the range of unachievable speed range for the rotor dynamic system. The full transient analysis of the rotor–motor AMB system is done analytically, and the exact voltage requires to pass through the resonance, while the system accelerated by the motor is obtained. From the steady-state analysis, it is observed that as the bias current is increased from 0 to 20 A, the jump voltage required to escape resonance has been decreased from 94.7 to 71.5 V. From frequency response curve, it is also observed that as the bias current is increased, it decreases the steady-state vibration amplitude up to 13.8% and a peculiar phenomenon like shifting of resonance is also occurred. It is also observed that percentage of reduction in the jump voltage required for passage through resonance are obtained as 4.5%, 17%, and 24% as the bias current supply to the AMB is increased from 0 to 5, 15 A, and 20 A, respectively, for transient analysis. Both the steady-state and transient responses were attained by increasing the bias current supply to the AMB system, and it is observed that the system vibration amplitude is reduced significantly at the resonance while increasing the bias current. The Sommerfeld effect characterization with respect to bias current supply to the AMB system is provided a vital idea of major reduction in the required jump voltage for passage through the resonance and achievable rotor speeds to operate the system in the post-resonance region.

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This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Authors and Affiliations

  1. Vibration and Rotor Dynamics Laboratory, Department of Mechanical Engineering, National Institute of Technology Silchar, Assam, 788010, India

    Prabina Kumar Meher, Mohd. Anis Ansari & Alfa Bisoi

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  1. Prabina Kumar Meher
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  2. Mohd. Anis Ansari
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  3. Alfa Bisoi
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Correspondence to Alfa Bisoi.

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Appendices

Appendix A

$${F}_{1}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}\gamma +y\mathrm{sin}\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}\gamma +y\mathrm{sin}\gamma \right)\right)}^{2}}\right]$$
(A1)
$${F}_{2}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}3\gamma +y\mathrm{sin}3\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}3\gamma +y\mathrm{sin}3\gamma \right)\right)}^{2}}\right]$$
(A2)
$${F}_{3}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}5\gamma +y\mathrm{sin}5\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}5\gamma +y\mathrm{sin}5\gamma \right)\right)}^{2}}\right]$$
(A3)
$${F}_{4}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}7\gamma +y\mathrm{sin}7\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}7\gamma +y\mathrm{sin}7\gamma \right)\right)}^{2}}\right]$$
(A4)
$${F}_{5}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}9\gamma +y\mathrm{sin}9\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}9\gamma +y\mathrm{sin}9\gamma \right)\right)}^{2}}\right]$$
(A5)
$${F}_{6}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}11\gamma +y\mathrm{sin}11\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}y+{K}_{d}\dot{y}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}11\gamma +y\mathrm{sin}11\gamma \right)\right)}^{2}}\right]$$
(A6)
$${F}_{7}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}13\gamma +y\mathrm{sin}13\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}13\gamma +y\mathrm{sin}13\gamma \right)\right)}^{2}}\right]$$
(A7)
$${F}_{8}={K}_{m}\left[\frac{\left({{I}_{0}-\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}-\left(x\mathrm{cos}15\gamma +y\mathrm{sin}15\gamma \right)\right)}^{2}}-\frac{\left({{I}_{0}+\left({K}_{p}x+{K}_{d}\dot{x}\right)}^{2}\right)}{{\left({S}_{0}+\left(x\mathrm{cos}15\gamma +y\mathrm{sin}15\gamma \right)\right)}^{2}}\right]$$
(A8)

Appendix B

$$\begin{aligned} \frac{\omega }{{2\pi }}\mathop \smallint \limits_{0}^{{2\pi /\omega }} D_{x} \dot{x}~dt & =0.5C{\omega }^{2}\left[{X}_{0}^{2}++2{X}_{0}{X}_{11}+{X}_{11}^{2}+9{X}_{13}^{2}\right]\\ &\quad+{0.125\beta }_{3}{\omega }^{2}\left[{X}_{0}^{4}+4{X}_{11}{X}_{0}^{3}+4{X}_{13}{A}_{0}^{3}+6{X}_{11}^{2}{X}_{0}^{2}+12{X}_{13}{X}_{11}{X}_{0}^{2}+20{X}_{13}^{2}{X}_{0}^{2}+4{X}_{0}{X}_{11}^{3} \right.\\ &\quad \left.+12{X}_{13}{X}_{0}{X}_{11}^{2}+40{A}_{0}{X}_{11}{X}_{13}^{2}+{X}_{11}^{4}+4{X}_{13}{X}_{11}^{3}+20{X}_{11}^{2}{X}_{13}^{2}+{9X}_{13}^{4}\right]\\ &\quad+{0.125\beta }_{4}{\omega }^{2}\left[3{X}_{0}^{4}+6{X}_{11}{X}_{0}^{3}-6{X}_{13}{X}_{0}^{3}+3{X}_{11}^{2}{X}_{0}^{2}-6{X}_{11}{X}_{13}{X}_{0}^{2}+18{X}_{13}^{2}{X}_{0}^{2}+{Y}_{11}^{2}{X}_{0}^{2}\right.\\ &\quad \left.-2{Y}_{13}{Y}_{11}{X}_{0}^{2}+2{Y}_{13}^{2}{X}_{0}^{2}+2{X}_{0}{X}_{11}{Y}_{11}^{2}-4{X}_{0}{X}_{11}{Y}_{11}{Y}_{13}+4{X}_{0}{X}_{11}{Y}_{13}^{2}+6{X}_{0}{X}_{13}{Y}_{11}^{2}\right.\\ &\quad \left.+{{X}_{11}^{2}Y}_{11}^{2}-2{X}_{11}^{2}{Y}_{11}{Y}_{13}+{{2X}_{11}^{2}Y}_{13}^{2}+6{X}_{11}{X}_{13}{Y}_{11}^{2}+{{18X}_{13}^{2}Y}_{11}^{2}+{{9X}_{13}^{2}Y}_{13}^{2}\right]\\ &\quad+{0.25\beta }_{5}{\omega }^{3}\left[{Y}_{11}{X}_{0}^{3}+3{Y}_{13}{X}_{0}^{3}+2{X}_{11}{Y}_{11}{X}_{0}^{2}+6{X}_{11}{Y}_{13}{X}_{0}^{2}+2{Y}_{11}{X}_{13}{X}_{0}^{2}+18{Y}_{13}{X}_{13}{X}_{0}^{2}\right.\\ &\quad \left.+{X}_{0}{Y}_{11}{X}_{11}^{2}+3{X}_{0}{Y}_{13}{X}_{11}^{2}+2{X}_{0}{X}_{11}{Y}_{11}{X}_{13}+18{X}_{0}{X}_{11}{X}_{13}{Y}_{13}\right]\\ &\quad-{0.125\beta }_{7}{\omega }^{2}\left[{X}_{0}^{4}+2{X}_{11}{X}_{0}^{3}+2{X}_{13}{X}_{0}^{3}+{X}_{11}^{2}{X}_{0}^{2}+2{X}_{11}{X}_{13}{X}_{0}^{2}-{Y}_{11}^{2}{X}_{0}^{2}-2{Y}_{11}{Y}_{13}{X}_{0}^{2}\right.\\ &\quad \left.-2{X}_{11}{X}_{0}{Y}_{11}^{2}-4{X}_{0}{X}_{11}{Y}_{11}{Y}_{13}-2{X}_{13}{X}_{0}{Y}_{11}^{2}-20{X}_{0}{Y}_{11}{X}_{13}{Y}_{13}-{X}_{11}^{2}{Y}_{11}^{2}\right.\\ &\quad \left.-2{Y}_{11}{Y}_{13}{X}_{11}^{2}-2{X}_{11}{X}_{13}{Y}_{11}^{2}-20{X}_{11}{Y}_{11}{Y}_{13}{X}_{13}-{{9X}_{13}^{2}Y}_{13}^{2}\right]\end{aligned}$$
(B1)
$$\begin{aligned} \frac{\omega }{{2\pi }}\mathop \smallint \limits_{0}^{{2\pi /\omega }}{D}_{x}\dot{y} dt&=0.5C{\omega }^{2}\left[{X}_{0}{Y}_{11}+{Y}_{11}{X}_{11}+9{X}_{13}{Y}_{13}\right]\\& \quad +{0.125\beta }_{3}{\omega }^{2}\left[{3Y}_{13}{X}_{0}^{3}+3{X}_{11}{Y}_{11}{X}_{0}^{2}+9{X}_{11}{Y}_{13}{X}_{0}^{2}+9{Y}_{11}{X}_{13}{X}_{0}^{2}+18{X}_{13}{Y}_{13}{X}_{0}^{2}\right. \\ &\quad \left.+3{X}_{0}{Y}_{11}{X}_{11}^{2}+9{X}_{0}{Y}_{13}{X}_{11}^{2}+2{X}_{11}{Y}_{11}{X}_{0}{X}_{13}+36{X}_{11}{X}_{0}{Y}_{13}{X}_{13}+2{X}_{0}{Y}_{11}{X}_{13}^{2}\right. \\ &\quad \left.+{Y}_{11}{X}_{11}^{3}+{3Y}_{13}{X}_{11}^{3}+{X}_{13}{Y}_{11}{X}_{11}^{2}+18{X}_{13}{Y}_{13}{X}_{11}^{2}+2{X}_{11}{Y}_{11}{X}_{13}^{2}+9{Y}_{13}{X}_{13}^{3}\right]\\& \quad +{0.125\beta }_{4}{\omega }^{2}\left[{Y}_{11}{X}_{0}^{3}-{Y}_{13}{X}_{0}^{3}+{Y}_{11}{X}_{11}{X}_{0}^{2}-{Y}_{13}{X}_{11}{X}_{0}^{2}-{9Y}_{11}{X}_{13}{X}_{0}^{2}+18{Y}_{13}{X}_{13}{X}_{0}^{2}\right. \\ &\quad \left.+{X}_{0}{Y}_{11}^{3}+{B}_{13}{X}_{0}{Y}_{11}^{2}+2{Y}_{11}{X}_{0}{Y}_{13}^{2}+{X}_{11}{Y}_{11}^{3}+{Y}_{13}{X}_{11}{Y}_{11}^{2}+{2Y}_{11}{X}_{11}{Y}_{13}^{2}+{3X}_{13}{Y}_{11}^{3}\right. \\ &\quad \left.+{18Y}_{13}{X}_{13}{Y}_{11}^{2}+{9X}_{13}{Y}_{13}^{3}\right]\\& \quad +{0.125\beta }_{5}{\omega }^{3}{A}_{0}\left[3{X}_{0}^{3}+3{X}_{11}{Y}_{0}^{2}+{X}_{13}{X}_{0}^{2}+3{X}_{0}{Y}_{11}^{2}+18{Y}_{11}{X}_{0}{Y}_{13}+{54X}_{0}{Y}_{13}^{2}\right. \\& \quad \left.+{3X}_{11}{Y}_{11}^{2}+18{Y}_{11}{X}_{11}{Y}_{13}+{54X}_{11}{Y}_{13}^{2}-{3X}_{13}{Y}_{11}^{2}\right]\\& \quad +{0.125\beta }_{6}{\omega }^{3}{X}_{0}\left[{X}_{0}^{3}+3{X}_{11}{X}_{0}^{2}+5{X}_{13}{X}_{0}^{2}+3{X}_{0}{X}_{11}^{2}+10{X}_{0}{X}_{11}{X}_{13}+{18X}_{0}{X}_{13}^{2}\right. \\ &\quad \left.+{X}_{11}^{3}+{5X}_{13}{X}_{11}^{2}+{18X}_{11}{X}_{13}^{2}\right]\\& \quad +{0.125\beta }_{7}{\omega }^{2}\left[{X}_{0}^{3}{Y}_{11}-{5X}_{0}^{3}{Y}_{13}+{Y}_{11}{X}_{11}{X}_{0}^{2}-5{Y}_{13}{X}_{11}{X}_{0}^{2}+{3Y}_{11}{X}_{13}{X}_{0}^{2}\right. \\ &\quad \left.+{2Y}_{13}{X}_{13}{X}_{0}^{2}+{X}_{0}{Y}_{11}^{3}+5{Y}_{13}{X}_{0}{Y}_{11}^{2}+18{Y}_{11}{X}_{0}{Y}_{13}^{2}+{X}_{11}{Y}_{11}^{3}+5{X}_{11}{Y}_{13}{Y}_{11}^{2}\right. \\ &\quad \left.+18{Y}_{11}{X}_{11}{Y}_{13}^{2}-{X}_{13}{Y}_{11}^{3}\right]\end{aligned}$$
(B2)
$$\begin{aligned} \frac{\omega }{{2\pi }}\mathop \smallint \limits_{0}^{{2\pi /\omega }} {D}_{y}\dot{x} dt&=0.5C{\omega }^{2}\left[{X}_{0}{Y}_{11}+{Y}_{11}{X}_{11}+9{X}_{13}{Y}_{13}\right]\\ & \quad+{0.125\beta }_{3}{\omega }^{2}\left[{Y}_{11}{X}_{0}^{3}-{Y}_{13}{X}_{0}^{3}+{Y}_{11}{X}_{11}{X}_{0}^{2}-{Y}_{13}{X}_{11}{X}_{0}^{2}-{9Y}_{11}{X}_{13}{X}_{0}^{2}+18{Y}_{13}{X}_{13}{X}_{0}^{2}\right. \\ & \quad \left.+{X}_{0}{Y}_{11}^{3}+{Y}_{13}{X}_{0}{Y}_{11}^{2}+2{Y}_{11}{X}_{0}{Y}_{13}^{2}+{X}_{11}{Y}_{11}^{3}+{Y}_{13}{X}_{11}{Y}_{11}^{2}+{2Y}_{11}{X}_{11}{Y}_{13}^{2}+{3X}_{13}{Y}_{11}^{3}\right. \\ & \quad \left.+{18Y}_{13}{X}_{13}{Y}_{11}^{2}+{9X}_{13}{Y}_{13}^{3}\right]\\ & \quad+{0.125\beta }_{4}{\omega }^{2}\left[{3Y}_{13}{X}_{0}^{3}+3{X}_{11}{Y}_{11}{X}_{0}^{2}+9{X}_{11}{Y}_{13}{X}_{0}^{2}+9{Y}_{11}{X}_{13}{X}_{0}^{2}+18{X}_{13}{Y}_{13}{X}_{0}^{2}\right. \\ & \quad \left.+3{X}_{0}{Y}_{11}{X}_{11}^{2}+9{X}_{0}{Y}_{13}{X}_{11}^{2}+2{X}_{11}{Y}_{11}{X}_{0}{X}_{13}+36{X}_{11}{X}_{0}{Y}_{13}{X}_{13}+2{X}_{0}{Y}_{11}{X}_{13}^{2}\right. \\ & \quad \left.+{Y}_{11}{X}_{11}^{3}+{3Y}_{13}{X}_{11}^{3}+{X}_{13}{Y}_{11}{X}_{11}^{2}+18{X}_{13}{Y}_{13}{X}_{11}^{2}+2{X}_{11}{Y}_{11}{X}_{13}^{2}+9{Y}_{13}{X}_{13}^{3}\right]\\ & \quad-{0.375\beta }_{5}{\omega }^{3}{X}_{0}\left[{X}_{0}^{3}+3{X}_{0}^{2}{X}_{11}-3{X}_{0}^{2}{X}_{13}+3{X}_{11}^{2}{X}_{0}-6{X}_{0}{X}_{11}{X}_{13}+18{X}_{13}^{2}{X}_{0}\right. \\ & \quad \left.+{X}_{11}^{3}-3{X}_{11}^{2}{X}_{13}+18{X}_{13}^{2}{X}_{11}-{0.125\beta }_{6}{\omega }^{3}{X}_{0}\left[{X}_{0}^{3}+{X}_{0}^{2}{X}_{11}+3{X}_{0}^{2}{X}_{13}+{Y}_{11}^{2}{X}_{0}\right. \right.\\ & \quad \left.\left.-10{X}_{0}{Y}_{11}{Y}_{13}+18{Y}_{13}^{2}{X}_{0}+{Y}_{11}^{2}{X}_{11}-10{X}_{11}{Y}_{11}{Y}_{13}+18{X}_{11}{X}_{13}^{2}-{9X}_{13}{Y}_{11}^{2}\right]\right.\\ & \quad \left.+{0.125\beta }_{7}{\omega }^{2}\left[{X}_{0}^{3}{Y}_{11}-{X}_{0}^{3}{Y}_{13}+3{X}_{11}{Y}_{11}{X}_{0}^{2}-3{X}_{11}{Y}_{13}{X}_{0}^{2}+5{X}_{13}{Y}_{11}{X}_{0}^{2}+2{X}_{13}{Y}_{13}{X}_{0}^{2}\right.\right. \\ & \quad \left.\left.+3{X}_{0}{Y}_{11}{X}_{11}^{2}-3{X}_{0}{Y}_{13}{X}_{11}^{2}+10{X}_{11}{X}_{0}{Y}_{11}{X}_{13}+4{X}_{11}{X}_{0}{Y}_{13}{X}_{13}+18{X}_{0}{Y}_{11}{X}_{13}^{2}\right.\right. \\ & \quad \left.\left.+{Y}_{11}{X}_{11}^{3}-{Y}_{13}{X}_{11}^{3}+5{X}_{13}{Y}_{11}{X}_{11}^{2}+2{X}_{13}{Y}_{13}{X}_{11}^{2}+18{X}_{11}{Y}_{11}{X}_{13}^{2}+{9Y}_{13}{X}_{13}^{2}\right]\right.\end{aligned}$$
(B3)
$$\begin{aligned} \frac{\omega }{{2\pi }}\mathop \smallint \limits_{0}^{{2\pi /\omega }} D_{y} \dot{y}~dt &=0.5C{\omega }^{2}\left[{X}_{0}^{2}+{Y}_{11}^{2}+9{Y}_{13}^{2}\right]\\ & \quad +{0.125\beta }_{3}{\omega }^{2}\left[{X}_{0}^{4}+{2X}_{0}^{2}{Y}_{11}^{2}-12{X}_{0}^{2}{Y}_{11}{Y}_{13}+{20X}_{0}^{2}{Y}_{13}^{2}+{Y}_{11}^{4}+4{Y}_{11}^{3}{Y}_{13}+{20Y}_{11}^{2}{Y}_{13}^{2}+9{Y}_{13}^{4}\right]\\ & \quad+{0.125\beta }_{4}{\omega }^{2}\left[{3X}_{0}^{4}+6{X}_{11}{X}_{0}^{3}+2{X}_{13}{X}_{0}^{3}+{3X}_{0}^{2}{X}_{11}^{2}+2{X}_{11}{X}_{13}{X}_{0}^{2}+{2X}_{0}^{2}{X}_{13}^{2}+{X}_{0}^{2}{Y}_{11}^{2}\right. \\ & \quad \left.+6{Y}_{11}{Y}_{13}{X}_{0}^{2}+18{X}_{0}^{2}{Y}_{13}^{2}+{2X}_{0}{X}_{11}{Y}_{11}^{2}+1{2X}_{0}{X}_{11}{Y}_{11}{Y}_{13}+3{6X}_{0}{X}_{11}{Y}_{13}^{2}\right. \\ & \quad \left.-2{X}_{0}{X}_{13}{Y}_{11}^{2}+{X}_{11}^{2}{Y}_{11}^{2}+6{X}_{11}^{2}{Y}_{11}{Y}_{13}+{18X}_{11}^{2}{Y}_{13}^{2}-2{X}_{11}{X}_{13}{Y}_{11}^{2}+2{X}_{13}^{2}{Y}_{11}^{2}{+9X}_{13}^{2}{Y}_{13}^{2}\right]\\ & \quad-{0.125\beta }_{5}{\omega }^{3}{X}_{0}\left[{X}_{0}^{2}{Y}_{11}-{X}_{0}^{2}{Y}_{13}+{2X}_{0}{X}_{11}{Y}_{11}-{2X}_{0}{X}_{11}{Y}_{13}-6{X}_{0}{X}_{13}{Y}_{11}\right. \\ & \quad \left.+1{8X}_{0}{X}_{13}{Y}_{13}+{X}_{11}^{2}{Y}_{11}-{X}_{11}^{2}{Y}_{13}-6{X}_{11}{Y}_{13}{Y}_{11}+18{X}_{11}{X}_{13}{Y}_{13}\right]\\ & \quad-{\beta }_{6}{\omega }^{3}{X}_{0}{Y}_{13}\left[{X}_{0}^{2}-3{Y}_{11}^{2}\right] \\ & \quad -{0.125\beta }_{7}{\omega }^{2}\left[{X}_{0}^{4}+2{X}_{11}{X}_{0}^{3}+2{X}_{13}{X}_{0}^{3}+{X}_{11}^{2}{X}_{0}^{2}+2{X}_{11}{X}_{13}{X}_{0}^{2}-{Y}_{11}^{2}{X}_{0}^{2}-2{Y}_{11}{Y}_{13}{X}_{0}^{2}\right. \\ & \quad \left.-2{X}_{11}{X}_{0}{Y}_{11}^{2}-4{X}_{0}{X}_{11}{Y}_{11}{Y}_{13}-2{X}_{13}{X}_{0}{Y}_{11}^{2}-20{X}_{0}{Y}_{11}{X}_{13}{Y}_{13}-{X}_{11}^{2}{Y}_{11}^{2}\right. \\ & \quad \left.-2{Y}_{11}{Y}_{13}{X}_{11}^{2}-2{X}_{11}{X}_{13}{Y}_{11}^{2}-20{X}_{11}{Y}_{11}{Y}_{13}{X}_{13}-{{9X}_{13}^{2}Y}_{13}^{2}\right] \end{aligned}$$
(B4)

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Meher, P.K., Ansari, M.A. & Bisoi, A. Effect of bias current of active magnetic bearing on Sommerfeld effect characterization in an unbalanced rotor dynamic system. Acta Mech 235, 907–923 (2024). https://doi.org/10.1007/s00707-023-03772-3

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