Investigation of the whirling motion and rub/impact occurrence in a 16-pole rotor active magnetic bearings system with constant stiffness

Abstract

In this paper, the weight of the rotor is considered in a 16-pole rotor active magnetic bearing system with constant stiffness. The equations of motion are derived to show the asymmetry between the rotor’s horizontal and vertical displacements. Accordingly, the rotor may exhibit forward, backward, intermediate, or hybrid whirls. The possibility to overcome the backward whirl and to symmetrize the rotor’s motion again is discussed. Also, the rotor may rub/impact with the stator legs depending on the values of the adopted system parameters. The multiple-scales method is utilized to extract the approximate solutions of the studied model and to analyze its nonlinear dynamics and the aforementioned whirls. The discussion is enhanced by different analytical plots such as the rotor’s responses to its eccentricity $f$ and rotation speed $\varOmega$. Numerical validation is carried out to demonstrate how these analytical plots describe precisely the nonlinear dynamical behavior of the whole system. Finally, whirl orbit maps are plotted to simulate the real-life motion of the rotor at different whirls and at rub/impact occurrence.

Appendices

Appendix A

Coefficients of Eq. (4):

$${\eta }_{1}=-\frac{16K{k}_{d}{I}_{0}}{{C}_{0}^{2}}$$

(A.1)

$${\eta }_{2}=\frac{16K}{{C}_{0}^{3}}\left[{I}_{0}^{2}-{k}_{p}{C}_{0}{I}_{0}\right]$$

(A.2)

$${\eta }_{3}=\frac{12K}{{C}_{0}^{5}}\left[2{I}_{0}^{2}-3{k}_{p}{C}_{0}{I}_{0}+{k}_{p}^{2}{C}_{0}^{2}\right]$$

(A.3)

$${\eta }_{4}=\frac{4K{k}_{d}}{{C}_{0}^{4}}\left[2{k}_{p}{C}_{0}-3{I}_{0}\right]$$

(A.4)

$${\eta }_{5}=\frac{4K{k}_{d}^{2}}{{C}_{0}^{3}}$$

(A.5)

Coefficients of Eq. (8):

$${\rho }_{11}=c-{\eta }_{1}-{\eta }_{4}{y}_{0}^{2}$$

(A.6)

$${\rho }_{12}=c-{\eta }_{1}-3{\eta }_{4}{y}_{0}^{2}$$

(A.7)

$${\rho }_{21}=-{\eta }_{2}-{\eta }_{3}{y}_{0}^{2}$$

(A.8)

$${\rho }_{22}=-{\eta }_{2}-3{\eta }_{3}{y}_{0}^{2}$$

(A.9)

Coefficients of Eq. (9):

$${\mu }_{1}=\frac{B{\rho }_{11}}{m}={c}_{1}+16d+4d{\gamma }^{2}[3-2p]$$

(A.10)

$${\mu }_{2}=\frac{B{\rho }_{12}}{m}={c}_{1}+16d+12d{\gamma }^{2}[3-2p]$$

(A.11)

$${\omega }_{1}^{2}=\frac{{B}^{2}{\rho }_{21}}{m}=16p-16+12{\gamma }^{2}\left[3p-{p}^{2}-2\right]$$

(A.12)

$${\omega }_{2}^{2}=\frac{{B}^{2}{\rho }_{22}}{m}=16p-16+36{\gamma }^{2}\left[3p-{p}^{2}-2\right]$$

(A.13)

$${\alpha }_{1}=-\frac{{B}^{2}{\eta }_{3}{\left({C}_{0}-{y}_{0}\right)}^{2}}{m}=12\left[3p-{p}^{2}-2\right]{[1-\gamma ]}^{2}$$

(A.14)

$${\alpha }_{2}=-\frac{B{\eta }_{4}{\left({C}_{0}-{y}_{0}\right)}^{2}}{m}=4d[3-2p]{[1-\gamma ]}^{2}$$

(A.15)

$${\alpha }_{3}=-\frac{{\eta }_{5}{\left({C}_{0}-{y}_{0}\right)}^{2}}{m}=-4{d}^{2}{[1-\gamma ]}^{2}$$

(A.16)

$${\alpha }_{4}=\frac{2{B}^{2}{\eta }_{3}{y}_{0}\left({C}_{0}-{y}_{0}\right)}{m}=-12\gamma \left[3p-{p}^{2}-2\right][1-\gamma ]$$

(A.17)

$${\alpha }_{5}=\frac{2B{\eta }_{4}{y}_{0}\left({C}_{0}-{y}_{0}\right)}{m}=-8d\gamma [3-2p][1-\gamma ]$$

(A.18)

$${\alpha }_{6}=\frac{2{\eta }_{5}{y}_{0}\left({C}_{0}-{y}_{0}\right)}{m}=4{d}^{2}\gamma [1-\gamma ]$$

(A.19)

$$p=\frac{{C}_{0}{k}_{p}}{{I}_{0}}$$

(A.20)

$$d=\frac{{C}_{0}{k}_{d}}{B{I}_{0}}$$

(A.21)

$${c}_{1}=\frac{Bc}{m}$$

(A.22)

$$f=\frac{e}{{C}_{0}-{y}_{0}}$$

(A.23)

$$\gamma =\frac{{y}_{0}}{{C}_{0}}$$

(A.24)

Coefficients of Eq. (19):

$$\begin{aligned} {\gamma }_{11}&=\left[\frac{3{\omega }_{2}^{2}-12{\omega }_{1}^{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{2}-\left[\frac{5{\omega }_{2}^{2}-8{\omega }_{1}^{2}}{8{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}{\alpha }_{5}\\&\quad -\left[\frac{{\omega }_{1}^{2}\left(5{\omega }_{2}^{2}-8{\omega }_{1}^{2}\right)}{8{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}{\alpha }_{6} \end{aligned}$$

(A.25)

$${\gamma }_{12}=-\frac{3{\alpha }_{2}}{8}+\frac{9{\alpha }_{4}{\alpha }_{5}}{8{\omega }_{2}^{2}}+\frac{9{\alpha }_{5}{\alpha }_{6}}{8}$$

(A.26)

$${\gamma }_{21}=-\frac{{\alpha }_{2}}{4}+\frac{3{\alpha }_{4}{\alpha }_{5}}{4{\omega }_{2}^{2}}+\frac{3{\alpha }_{5}{\alpha }_{6}}{4}$$

(A.27)

$$\begin{aligned}{\gamma }_{22}&=-\frac{{\alpha }_{2}}{4}+\left[\frac{8{\omega }_{1}^{2}+{\omega }_{2}^{2}}{4{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}{\alpha }_{5}\\&\quad +\left[\frac{{\omega }_{1}^{2}\left(8{\omega }_{1}^{2}+{\omega }_{2}^{2}\right)}{4{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}{\alpha }_{6}\end{aligned}$$

(A.28)

$$\begin{aligned}{\gamma }_{31}&=-\frac{{\alpha }_{1}}{8{\omega }_{1}}-\left[\frac{{\omega }_{2}^{3}-4{\omega }_{2}^{2}{\omega }_{1}+4{\omega }_{1}^{2}{\omega }_{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{1}}\right]{\alpha }_{3}-\left[\frac{4{\omega }_{1}-6{\omega }_{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{2}^{2}{\omega }_{1}}\right]{\alpha }_{4}^{2}\\&\quad -\left[\frac{6{\omega }_{2}^{2}-16{\omega }_{2}{\omega }_{1}+8{\omega }_{1}^{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{2}{\omega }_{1}}\right]{\alpha }_{4}{\alpha }_{6}-\left[\frac{3{\omega }_{1}^{2}-8{\omega }_{2}{\omega }_{1}+4{\omega }_{2}^{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{2}{\omega }_{1}}\right]{\alpha }_{5}^{2}-\left[\frac{\left(8{\omega }_{2}-12{\omega }_{1}\right){\omega }_{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right)}\right]{\alpha }_{6}^{2}\end{aligned}$$

(A.29)

$$\begin{aligned}{\gamma }_{32}&=\frac{{\alpha }_{1}}{8{\omega }_{2}}-\left[\frac{{\omega }_{1}\left({\omega }_{1}-2{\omega }_{2}\right)}{8{\omega }_{2}}\right]{\alpha }_{3}+\left[\frac{\left({\omega }_{2}-4{\omega }_{1}\right)}{4{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}^{2}\\&\quad +\left[\frac{{\omega }_{1}\left(2{\omega }_{2}^{2}-9{\omega }_{1}{\omega }_{2}+4{\omega }_{1}^{2}\right)}{4{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}{\alpha }_{6}+\left[\frac{2{\omega }_{1}^{2}-4{\omega }_{1}{\omega }_{2}-{\omega }_{2}^{2}}{8{\omega }_{2}^{2}\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}^{2}+\left[\frac{{\omega }_{1}^{2}\left(2{\omega }_{1}^{2}-4{\omega }_{1}{\omega }_{2}-{\omega }_{2}^{2}\right)}{2{\omega }_{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}^{2}\end{aligned}$$

(A.30)

$$\begin{aligned}{\gamma }_{41}&=\left[\frac{2{\omega }_{1}^{2}+2{\omega }_{2}^{2}-5{\omega }_{1}{\omega }_{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{1}}\right]{\alpha }_{2}+\left[\frac{-13{\omega }_{1}{\omega }_{2}+2{\omega }_{1}^{2}+10{\omega }_{2}^{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{2}^{2}{\omega }_{1}}\right]{\alpha }_{4}{\alpha }_{5}\\&\quad +\left[\frac{-14{\omega }_{1}^{2}+17{\omega }_{1}{\omega }_{2}-4{\omega }_{2}^{2}}{8\left(2{\omega }_{1}-{\omega }_{2}\right){\omega }_{1}}\right]{\alpha }_{5}{\alpha }_{6}\end{aligned}$$

(A.31)

$$\begin{aligned}{\gamma }_{42}&=-\left[\frac{2{\omega }_{1}-{\omega }_{2}}{8{\omega }_{2}}\right]{\alpha }_{2}+\left[\frac{12{\omega }_{1}^{2}-14{\omega }_{1}{\omega }_{2}-{\omega }_{2}^{2}}{8{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}{\alpha }_{5}\\&\quad -\left[\frac{{\omega }_{1}\left({\omega }_{1}-4{\omega }_{2}\right)\left(4{\omega }_{1}^{2}-4{\omega }_{1}{\omega }_{2}-{\omega }_{2}^{2}\right)}{8{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}{\alpha }_{6}\end{aligned}$$

(A.32)

$$\begin{aligned}{\gamma }_{51}&=-\left[\frac{-3{\omega }_{2}^{2}+12{\omega }_{1}^{2}}{8{\omega }_{1}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{1}-\left[\frac{12{\omega }_{1}^{3}-3{\omega }_{2}^{2}{\omega }_{1}}{8\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{3}\\&\quad -\left[\frac{6{\omega }_{2}^{2}-16{\omega }_{1}^{2}}{8{\omega }_{2}^{2}{\omega }_{1}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}^{2}-\left[\frac{6{\omega }_{1}{\omega }_{2}^{2}-16{\omega }_{1}^{3}}{8{\omega }_{2}^{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}{\alpha }_{4}\\&\quad +\left[\frac{{\omega }_{1}}{8\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}^{2}+\left[\frac{{\omega }_{1}^{3}}{2\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}^{2}\end{aligned}$$

(A.33)

$${\gamma }_{52}=-\frac{3{\alpha }_{1}}{8{\omega }_{2}}+\frac{15{\alpha }_{4}^{2}}{4{\omega }_{2}^{3}}+\frac{15{\alpha }_{4}{\alpha }_{6}}{4{\omega }_{2}}+\frac{3{\alpha }_{6}^{2}{\omega }_{2}}{2}-\frac{3{\alpha }_{3}{\omega }_{2}}{8}+\frac{3{\alpha }_{5}^{2}}{8{\omega }_{2}}$$

(A.34)

$$\begin{aligned}{\gamma }_{61}&=-\frac{{\alpha }_{1}}{4{\omega }_{1}}-\left[\frac{-{\omega }_{2}^{3}+4{\omega }_{2}^{2}{\omega }_{1}^{2}}{4{\omega }_{1}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{3}-\left[\frac{-24{\omega }_{1}^{2}+2{\omega }_{2}^{2}}{4{\omega }_{2}^{2}{\omega }_{1}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}^{2}\\&\quad -\left[\frac{2{\omega }_{2}^{2}-24{\omega }_{1}^{2}}{4{\omega }_{1}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}{\alpha }_{4}+\left[\frac{{\omega }_{1}}{4\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}^{2}+\left[\frac{{\omega }_{2}^{2}{\omega }_{1}}{\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}^{2}\end{aligned}$$

(A.35)

$$\begin{aligned}{\gamma }_{62}&=-\frac{{\alpha }_{1}}{4{\omega }_{2}}-\frac{{\alpha }_{3}{\omega }_{1}^{2}}{4{\omega }_{2}}+\left[\frac{\left(12{\omega }_{1}^{2}-{\omega }_{2}^{2}\right)}{2{\omega }_{2}^{3}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}^{2}+\left[\frac{{\omega }_{1}^{2}\left(12{\omega }_{1}^{2}-{\omega }_{2}^{2}\right)}{2{\omega }_{2}^{3}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{4}{\alpha }_{6}\\&\quad +\left[\frac{2{\omega }_{1}^{2}-{\omega }_{2}^{2}}{4{\omega }_{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{5}^{2}+\left[\frac{{\omega }_{1}^{2}\left(2{\omega }_{1}^{2}-{\omega }_{2}^{2}\right)}{{\omega }_{2}\left(2{\omega }_{1}-{\omega }_{2}\right)\left(2{\omega }_{1}+{\omega }_{2}\right)}\right]{\alpha }_{6}^{2}\end{aligned}$$

(A.36)

Appendix B

$${J}_{11}=\frac{\partial {H}_{1}}{\partial {a}_{1}}=-\frac{{\mu }_{1}}{2}+3{\gamma }_{11}{a}_{1}^{2}+{\gamma }_{21}{a}_{2}^{2}+{\gamma }_{31}{a}_{2}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+{\gamma }_{41}{a}_{2}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.1)

$${J}_{12}=\frac{\partial {H}_{1}}{\partial {a}_{2}}=2{\gamma }_{21}{a}_{1}{a}_{2}+2{\gamma }_{31}{a}_{1}{a}_{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{41}{a}_{1}{a}_{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.2)

$${J}_{13}=\frac{\partial {H}_{1}}{\partial {\phi }_{1}}=2{\gamma }_{31}{a}_{1}{a}_{2}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-2{\gamma }_{41}{a}_{1}{a}_{2}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+\frac{f{\varOmega }^{2}}{2{\omega }_{1}}{\mathrm{cos}}\left({\phi }_{1}\right)$$

(B.3)

$${J}_{14}=\frac{\partial {H}_{1}}{\partial {\phi }_{2}}=-2{\gamma }_{31}{a}_{1}{a}_{2}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{41}{a}_{1}{a}_{2}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.4)

$${J}_{21}=\frac{\partial {H}_{2}}{\partial {a}_{1}}=2{\gamma }_{22}{a}_{1}{a}_{2}+2{\gamma }_{32}{a}_{1}{a}_{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{42}{a}_{1}{a}_{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.5)

$${J}_{22}=\frac{\partial {H}_{2}}{\partial {a}_{2}}=-\frac{{\mu }_{2}}{2}+3{\gamma }_{12}{a}_{2}^{2}+{\gamma }_{22}{a}_{1}^{2}+{\gamma }_{32}{a}_{1}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+{\gamma }_{42}{a}_{1}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.6)

$${J}_{23}=\frac{\partial {H}_{2}}{\partial {\phi }_{1}}=2{\gamma }_{32}{a}_{1}^{2}{a}_{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-2{\gamma }_{42}{a}_{1}^{2}{a}_{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.7)

$${J}_{24}=\frac{\partial {H}_{2}}{\partial {\phi }_{2}}=-2{\gamma }_{32}{a}_{1}^{2}{a}_{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{42}{a}_{1}^{2}{a}_{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+\frac{f{\varOmega }^{2}}{2{\omega }_{2}}{\mathrm{sin}}\left({\phi }_{2}\right)$$

(B.8)

$${J}_{31}=\frac{\partial {H}_{3}}{\partial {a}_{1}}=2{\gamma }_{51}{a}_{1}-\frac{f{\varOmega }^{2}}{2{\omega }_{1}}\frac{{\mathrm{cos}}\left({\phi }_{1}\right)}{{a}_{1}^{2}}$$

(B.9)

$${J}_{32}=\frac{\partial {H}_{3}}{\partial {a}_{2}}=2{\gamma }_{61}{a}_{2}+2{\gamma }_{31}{a}_{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-2{\gamma }_{41}{a}_{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.10)

$${J}_{33}=\frac{\partial {G}_{3}}{\partial {\phi }_{1}}=-2{\gamma }_{31}{a}_{2}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-2{\gamma }_{41}{a}_{2}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-\frac{f{\varOmega }^{2}}{2{\omega }_{1}}\frac{{\mathrm{sin}}\left({\phi }_{1}\right)}{{a}_{1}}$$

(B.11)

$${J}_{34}=\frac{\partial {H}_{3}}{\partial {\phi }_{2}}=2{\gamma }_{31}{a}_{2}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{41}{a}_{2}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.12)

$${J}_{41}=\frac{\partial {H}_{4}}{\partial {a}_{1}}=2{\gamma }_{62}{a}_{1}-2{\gamma }_{32}{a}_{1}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{42}{a}_{1}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.13)

$${J}_{42}=\frac{\partial {H}_{4}}{\partial {a}_{2}}=2{\gamma }_{52}{a}_{2}-\frac{f{\varOmega }^{2}}{2{\omega }_{2}}\frac{{\mathrm{sin}}\left({\phi }_{2}\right)}{{a}_{2}^{2}}$$

(B.14)

$${J}_{43}=\frac{\partial {H}_{4}}{\partial {\phi }_{1}}=2{\gamma }_{32}{a}_{1}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+2{\gamma }_{42}{a}_{1}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)$$

(B.15)

$${J}_{44}=\frac{\partial {H}_{4}}{\partial {\phi }_{2}}=2{\gamma }_{32}{a}_{1}^{2}{\mathrm{sin}}\left(2{\phi }_{1}-2{\phi }_{2}\right)-2{\gamma }_{42}{a}_{1}^{2}{\mathrm{cos}}\left(2{\phi }_{1}-2{\phi }_{2}\right)+\frac{f{\varOmega }^{2}}{2{\omega }_{2}}\frac{{\mathrm{cos}}\left({\phi }_{2}\right)}{{a}_{2}}$$

(B.16)