Appendix
$$\begin{aligned} J_{11}= & {} -\frac{\mu }{2}+\frac{\alpha _{2} -\alpha _{5} \omega ^{2}}{8\omega }a_{20}^{2} \sin 2\phi _{20} -\frac{\beta _{3} }{8\omega }a_{30}^{2} \sin 2\phi _{30} \\&+\,\frac{\beta _{5} }{8\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) -\frac{\beta _{1} }{4\omega }a_{10} a_{30} \sin \phi _{30} \\&+\,\frac{\beta _{4} }{8\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\alpha _{4} }{8}a_{20}^{2} \cos 2\phi _{20} -\frac{\alpha _{7} }{8}a_{20}^{2} \cos 2\phi _{20} \\&-\,\frac{\beta _{2} }{4}a_{10} a_{30} \cos \phi _{30} -\frac{\beta _{6} }{8}a_{20}a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{3\alpha _{3} }{8}a_{10}^{2} -\frac{\alpha _{4} }{4}a_{20}^{2} \\ J_{12}= & {} \frac{f\Omega ^{2}}{2\omega }\cos \phi _{10}\\ J_{13}= & {} \frac{\alpha _{2} -\alpha _{5} \omega ^{2}}{4\omega }a_{10} a_{20} \sin 2\phi _{20} -\frac{\beta _{7} }{2\omega }a_{20} a_{30} \sin \phi _{30} \\&+\,\frac{\beta _{4} }{8\omega }a_{10} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{7} }{4\omega }a_{20} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\alpha _{4} }{4}a_{10} a_{20} \cos 2\phi _{20} -\frac{\alpha _{7} }{4}a_{10} a_{20} \cos 2\phi _{20} \\&-\,\frac{\beta _{6} }{8}a_{1} a_{4} \cos \left( {2\phi _{20} -\phi _{40} } \right) -\frac{\alpha _{4} }{2}a_{10} a_{20} \\ \end{aligned}$$
$$\begin{aligned} J_{14}= & {} \frac{\alpha _{2} -\alpha _{5} \omega ^{2}}{4\omega }a_{10} a_{20}^{2} \cos 2\phi _{20} \\&+\,\frac{\beta _{5} }{4\omega }a_{10} a_{40}^{2} \cos \left( {2\phi _{20} -2\phi _{40} } \right) \\&+\,\frac{\beta _{4} }{4\omega }a_{10} a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{7} }{4\omega }a_{20}^{2} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\alpha _{4} }{4}a_{10} a_{20}^{2} \sin 2\phi _{20} +\frac{\alpha _{7} }{4}a_{10} a_{20}^{2} \sin 2\phi _{20} \\&+\,\frac{\beta _{6} }{4}a_{10} a_{20} a_{40} \sin \left( {2\phi _{20}-\phi _{40} } \right) \\ J_{15}= & {} -\frac{\beta _{3} }{4\omega }a_{10} a_{30} \sin 2\phi _{30} +\frac{\eta _{1} }{2\omega }\sin \phi _{30} \\&-\,\frac{\beta _{7} }{4\omega }a_{20}^{2} \sin \phi _{30} -\frac{\beta _{1} }{8\omega }a_{10}^{2} \sin \phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }a_{20}^{2} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\beta _{2} }{8}a_{10}^{2} \cos \phi _{30} \\ J_{16}= & {} -\frac{\beta _{3} }{4\omega }a_{10} a_{30}^{2} \cos 2\phi _{30} +\frac{\eta _{1} }{2\omega }a_{30} \cos \phi _{30}\\&-\,\frac{\beta _{7} }{4\omega }a_{20}^{2} a_{30} \cos \phi _{30} -\frac{\beta _{1} }{8\omega }a_{10}^{2} a_{30} \cos \phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }a_{20}^{2} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\beta _{2} }{8}a_{10}^{2} a_{30} \sin \phi _{30} \\ J_{17}= & {} \frac{\beta _{5} }{4\omega }a_{10} a_{40} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\&+\,\frac{\beta _{4} }{8\omega }a_{10} a_{20} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{6} }{8}a_{10} a_{20} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\ J_{18}= & {} -\frac{\beta _{5} }{4\omega }a_{10} a_{40}^{2} \cos \left( {2\phi _{20} -2\phi _{40} } \right) \\&-\,\frac{\beta _{4} }{8\omega }a_{10} a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\beta _{6} }{8}a_{10} a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\ J_{21}= & {} -\frac{\beta _{2} }{8}a_{30} \sin \phi _{30} -\frac{\beta _{6} }{4}a_{20} a_{40} \sin \phi _{40} \\&-\,\frac{3\beta _{1} }{8\omega }a_{30} \cos \phi _{30} -\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10}^{2} }\cos \phi _{30} \\&+\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{3\alpha _{1} }{4\omega }a_{10} -\frac{\alpha _{6} \omega }{4}a_{10} -\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{10}^{2} }\cos \phi _{10} \\ J_{22}= & {} -\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{10} }\sin \phi _{10}\\ J_{23}= & {} \frac{\alpha _{5} \omega }{4}a_{20} \cos 2\phi _{20} -\frac{\alpha _{2} }{4\omega }a_{20} \cos 2\phi _{20} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\beta _{6} }{4}a_{40} \sin \phi _{40} +\frac{\alpha _{4} }{4}a_{20} \sin 2\phi _{20}\\&-\,\frac{\alpha _{7}}{4}a_{20} \sin 2\phi _{20} -\frac{\beta _{6} }{8}a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{7} }{2\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \phi _{30} -\frac{\beta _{4} }{4\omega }a_{40} \cos \phi _{40}\\&-\,\frac{\beta _{7} }{4\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\beta _{4} }{8\omega }a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) -\frac{\alpha _{2} }{2\omega }a_{20} -\frac{\omega \alpha _{5} }{2}a_{20} \\ J_{24}= & {} -\frac{\alpha _{5} \omega }{4}a_{20}^{2} \sin 2\phi _{20} +\frac{\alpha _{2} }{4\omega }a_{20}^{2} \sin 2\phi _{20}\\&+\,\frac{\alpha _{4} }{4}a_{20}^{2} \cos 2\phi _{20} -\frac{\alpha _{7} }{4}a_{20}^{2} \cos 2\phi _{20} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\beta _{6} }{4}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40}} \right) \\&+\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\ J_{25}= & {} -\frac{\beta _{2} }{8}a_{10} \sin \phi _{30} -\frac{3\beta _{1} }{8\omega }a_{10} \cos \phi _{30} \\&+\,\frac{\eta _{1} }{2\omega a_{10} }\cos \phi _{30} -\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} }{a_{10} }\cos \phi _{30}\\&-\frac{\beta _{3} }{4\omega }a_{30} \cos 2\phi _{30} \\&-\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} } \right) -\frac{\beta _{3} }{2\omega }a_{30} \\ J_{26}= & {} -\frac{\beta _{2} }{8}a_{10} a_{30} \cos \phi _{30} +\frac{3\beta _{1} }{8\omega }a_{10} a_{30} \sin \phi _{30}\\&-\,\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10} }\sin \phi _{30} +\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \phi _{30} \\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\beta _{3} }{4\omega }a_{30}^{2} \sin 2\phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\ J_{27}= & {} -\frac{\beta _{6} }{4}a_{20} \sin \phi _{40} -\frac{\beta _{6} }{8}a_{20} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{4} }{4\omega }a_{20} \cos \phi _{40} -\frac{\beta _{4} }{8\omega }a_{20} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{5} }{4\omega }a_{40} \cos \left( {2\phi _{20} -2\phi _{40} } \right) -\frac{\beta _{5} }{2\omega }a_{40} \\ J_{28}= & {} -\frac{\beta _{6} }{4}a_{20} a_{40} \cos \phi _{40} +\frac{\beta _{6} }{8}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \phi _{40} -\frac{\beta _{4} }{8\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\ J_{31}= & {} -\frac{\alpha _{2} }{4\omega }a_{10} a_{20} \sin 2\phi _{20} +\frac{\alpha _{5} \omega }{4}a_{10} a_{20} \sin 2\phi _{20}\\&-\,\frac{\gamma _{7} }{2\omega }a_{10} a_{40} \sin \phi _{40}\\&-\,\frac{\gamma _{7} }{4\omega }a_{10} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{4} }{8\omega }a_{20} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\alpha _{4} }{4}a_{10} a_{20} \cos 2\phi _{20} -\frac{\alpha _{7} }{4}a_{10} a_{20} \cos 2\phi _{20}\\&-\,\frac{\gamma _{6} }{8}a_{20} a_{30} \cos \left( {2\phi _{20} +\phi _{30} }\right) -\frac{\alpha _{4} }{2}a_{10} a_{20} \\ J_{32}= & {} \frac{f\Omega ^{2}}{2\omega }\sin \left( {\phi _{10} +\phi _{20} } \right) \\ \end{aligned}$$
$$\begin{aligned} J_{33}= & {} -\frac{\mu }{2}-\frac{\alpha _{2} }{8\omega }a_{10}^{2} \sin 2\phi _{20} +\frac{\alpha _{5} \omega }{8}a_{10}^{2} \sin 2\phi _{20}\\&-\,\frac{\gamma _{5} }{8\omega }a_{30}^{2} \sin \left( {2\phi _{20} +2\phi _{30} }\right) -\frac{\gamma _{3} }{8\omega }a_{40}^{2} \sin 2\phi _{40}\\&-\,\frac{\gamma _{1} }{4\omega }a_{20} a_{40} \sin \phi _{40}\\&-\,\frac{\gamma _{4} }{8\omega }a_{10} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\alpha _{4} }{8}a_{10}^{2} \cos 2\phi _{20} -\frac{\alpha _{7} }{8}a_{10}^{2} \cos 2\phi _{20}\\&-\,\frac{\gamma _{2} }{4}a_{20} a_{40} \cos \phi _{40} \\&-\,\frac{\gamma _{6} }{8}a_{10} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{3\alpha _{3} }{8}a_{20}^{2} -\frac{\alpha _{4} }{4}a_{10}^{2} \\ J_{34}= & {} -\frac{\alpha _{2} }{4\omega }a_{10}^{2} a_{20} \cos 2\phi _{20} +\frac{\alpha _{5} \omega }{4}a_{10}^{2} a_{20} \cos 2\phi _{20} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{5} }{4\omega }a_{20} a_{30}^{2} \cos \left( {2\phi _{20} +2\phi _{30} } \right) \\&-\,\frac{\gamma _{7} }{4\omega }a_{10}^{2} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{4} }{4\omega }a_{10} a_{20} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\alpha _{4} }{4}a_{10}^{2} a_{20} \sin 2\phi _{20}\\&+\frac{\alpha _{7} }{4}a_{10}^{2} a_{20} \sin 2\phi _{20} \\&+\,\frac{\gamma _{6} }{4}a_{10} a_{20} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{f\Omega ^{2}}{2\omega }\sin \left( {\phi _{10} +\phi _{20} } \right) \\ J_{35}= & {} -\frac{\gamma _{5} }{4\omega }a_{20} a_{30} \sin \left( {2\phi _{20} +2\phi _{30} } \right) \\&-\,\frac{\gamma _{4} }{8\omega }a_{10} a_{20} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{6} }{8}a_{10} a_{20} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\ J_{36}= & {} -\frac{\gamma _{5} }{4\omega }a_{20} a_{30}^{2} \cos \left( {2\phi _{20} +2\phi _{30} } \right) \\&-\,\frac{\gamma _{4} }{8\omega }a_{10} a_{20} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{6} }{8}a_{10} a_{20} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\ J_{37}= & {} -\frac{\gamma _{3} }{4\omega }a_{20} a_{40} \sin 2\phi _{40} +\frac{\eta _{2} }{2\omega }\sin \phi _{40} \\&-\,\frac{\gamma _{1} }{8\omega }a_{20}^{2} \sin \phi _{40} \\&-\,\frac{\gamma _{7} }{4\omega }a_{10}^{2} \sin \phi _{40} -\frac{\gamma _{7} }{8\omega }a_{10}^{2} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{2} }{8}a_{20}^{2} \cos \phi _{40} \\ J_{38}= & {} -\frac{\gamma _{3} }{4\omega }a_{20} a_{40}^{2} \cos 2\phi _{40} +\frac{\eta _{2} }{2\omega }a_{40} \cos \phi _{40}\\&-\,\frac{\gamma _{1} }{8\omega }a_{20}^{2} a_{40} \cos \phi _{40} -\frac{\gamma _{7} }{4\omega }a_{10}^{2} a_{40} \cos \phi _{40} \\&-\,\frac{\gamma _{7} }{8\omega }a_{10}^{2} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) +\frac{\gamma _{2} }{8}a_{20}^{2} a_{40} \sin \phi _{40} \\ \end{aligned}$$
$$\begin{aligned} J_{41}= & {} \frac{\beta _{2} }{8}a_{30} \sin \phi _{30} -\frac{\gamma _{7} }{4\omega }\frac{a_{10} a_{40} }{a_{20} }\cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \phi _{30} +\frac{3\beta _{1} }{8\omega }a_{30} \cos \phi _{30}\\&+\,\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10}^{2} }\cos \phi _{30} \\&+\,\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{10}^{2} }\cos \phi _{10} +\frac{3\alpha _{1} }{4\omega }a_{10} \\&+\,\frac{\alpha _{6} \omega }{4}a_{10} -\frac{\gamma _{7} }{2\omega }\frac{a_{10} a_{40} }{a_{20} }\cos \phi _{40} -\frac{\alpha _{2} }{2\omega }a_{10} \\&-\,\frac{\alpha _{5} \omega }{2}a_{10} -\frac{\alpha _{4} }{4}a_{10} \sin 2\phi _{20} \\&+\,\frac{\alpha _{7} }{4}a_{10} \sin 2\phi _{20} -\frac{\gamma _{4} }{4\omega }a_{30} \cos \phi _{30} \\&-\,\frac{\gamma _{4} }{8\omega }a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\gamma _{6} }{4}a_{30} \sin \phi _{30} +\frac{\gamma _{6} }{8}a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\alpha _{2} }{4\omega }a_{10} \cos 2\phi _{20} +\frac{\alpha _{5} \omega }{4}a_{10} \cos 2\phi _{20} \\ \quad J_{42}= & {} \frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{10} }\sin \phi _{10} \\&+\,\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20} }\cos \left( {\phi _{10} +\phi _{20} } \right) \\ J_{43}= & {} \frac{\beta _{6} }{4}a_{40} \sin \phi _{40} +\frac{\beta _{6} }{8}a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} a_{40} }{a_{20}^{2} }\cos \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{7} }{2\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \phi _{30} \\&+\,\frac{\alpha _{2} }{4\omega }a_{20} \cos 2\phi _{20} -\frac{\alpha _{5} \omega }{4}a_{20} \cos 2\phi _{20}\\&+\,\frac{\beta _{4} }{4\omega }a_{40} \cos \phi _{40} +\frac{\beta _{4} }{8\omega }a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\alpha _{2} }{2\omega }a_{20} +\frac{\alpha _{5} \omega }{2}a_{20} \\&-\,\frac{\alpha _{4} }{4}a_{20} \sin 2\phi _{20} +\frac{\alpha _{7} }{4}a_{20} \sin 2\phi _{20} \\&+\,\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40} }{a_{20}^{2} }\cos \phi _{40} \\&-\,\frac{\gamma _{2} }{8}a_{40} \sin \phi _{40} +\frac{\beta _{7} }{4\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{3\gamma _{1} }{8\omega }a_{40} \cos \phi _{40} -\frac{\eta _{2} }{2\omega }\frac{a_{40} }{a_{20}^{2} }\cos \phi _{40}\\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{3\alpha _{1} }{4\omega }a_{20} -\frac{\alpha _{6} \omega }{4}a_{20} -\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20}^{2} }\sin \left( {\phi _{10} +\phi _{20} } \right) \\ J_{44}= & {} -\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\&+\,\frac{\beta _{6} }{4}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40}}{a_{20} }\sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\alpha _{2} }{4\omega }a_{20}^{2} \sin 2\phi _{20} +\frac{\alpha _{5} \omega }{4}a_{20}^{2} \sin 2\phi _{20}\\&-\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\alpha _{4} }{4}a_{20}^{2} \cos 2\phi _{20} +\frac{\alpha _{7} }{4}a_{20}^{2} \cos 2\phi _{20} \\&-\,\frac{\alpha _{4} }{4}a_{10}^{2} \cos 2\phi _{20} +\frac{\alpha _{7} }{4}a_{10}^{2} \cos 2\phi _{20} \\&+\,\frac{\gamma _{4} }{4\omega }a_{10} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{6} }{4}a_{10} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\gamma _{5} }{4\omega }a_{30}^{2} \sin \left( {2\phi _{20} +2\phi _{30} } \right) \\&-\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\alpha _{2} }{4\omega _{40} -\frac{\beta _{6} }{8}a_{20} \sin \left( {2\phi _{20} -\phi _{40} } \right) -\frac{\beta _{4} }{4\omega }a_{20} }\\&\times a_{10}^{2} \sin 2\phi _{20} \\&-\,\frac{\alpha _{5} \omega }{4}a_{10}^{2} \sin 2\phi _{20} \\&+\,\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20} }\cos \left( {\phi _{10} +\phi _{20} } \right) \\ J_{45}= & {} \frac{\beta _{2} }{8}a_{10} \sin \phi _{30} +\frac{\beta _{3} }{4\omega }a_{30} \cos 2\phi _{30} \\&+\,\frac{3\beta _{1} }{8\omega }a_{10} \cos \phi _{30} -\frac{\eta _{1} }{2\omega a_{10} }\cos \phi _{30} \\&+\,\frac{\beta _{3} }{2\omega }a_{30} +\frac{\beta _{5} }{4\omega }a_{40}^{2} \\&-\,\frac{\gamma _{5} }{2\omega }a_{30} -\frac{\gamma _{4} }{4\omega }a_{10} \cos \phi _{30} \\&-\,\frac{\gamma _{4} }{8\omega }a_{10} \cos \left( {2\phi _{20} +\phi _{30} } \right) -\frac{\gamma _{6} }{4}a_{10} \sin \phi _{30} \\&+\,\frac{\gamma _{6} }{8}a_{10} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{5} }{4\omega }a_{30} \cos \left( {2\phi _{20} +2\phi _{30} } \right) \\&+\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\ J_{46}= & {} \frac{\beta _{2} }{8}a_{10} a_{30} \cos \phi _{30} -\frac{\beta _{3} }{4\omega }a_{30}^{2} \sin 2\phi _{30}\\&-\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \phi _{30} -\frac{3\beta _{1} }{8\omega }a_{10} a_{30} \sin \phi _{30}\\&+\,\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10} }\sin \phi _{30} \\&+\,\frac{\gamma _{4} }{4\omega }a_{10} a_{30} \sin \phi _{30}\\&+\,\frac{\gamma _{4} }{8\omega }a_{10} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\gamma _{6} }{4}a_{10} a_{30} \cos \phi _{30}\\&+\,\frac{\gamma _{6} }{8}a_{10} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{5} }{4\omega }a_{30}^{2} \sin \left( {2\phi _{20} +2\phi _{30} } \right) \\&-\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned} J_{47}= & {} \frac{\beta _{6} }{4}a_{20} \sin \phi _{40} +\frac{\beta _{5} }{4\omega }a_{40} \cos \left( {2\phi _{20} -2\phi _{40} } \right) \\&+\,\frac{\beta _{6} }{8}a_{20} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} }{a_{20} }\cos \left( {2\phi _{20}-\phi _{40} } \right) \\&+\,\frac{\beta _{5} }{2\omega }a_{40} +\frac{\beta _{4} }{4\omega }a_{20} \cos \phi _{40} \\&+\,\frac{\beta _{4} }{8\omega }a_{20} \cos \left( {2\phi _{20} -\phi _{40} } \right) -\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} }{a_{20} }\cos \phi _{40} \\&-\,\frac{\gamma _{3} }{2\omega }a_{40}-\frac{\gamma _{2} }{8}a_{20}\sin \phi _{40} \\&-\,\frac{\gamma _{3} }{4\omega }a_{40} \cos 2\phi _{40}\\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{3\gamma _{1} }{8\omega }a_{20} \cos \phi _{40} +\frac{\eta _{2}}{2\omega a_{20} }\cos \phi _{40} \\ J_{48}= & {} \frac{\beta _{6} }{4}a_{20} a_{40} \cos \phi _{40}\\&+\,\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\&-\,\frac{\beta _{6} }{8}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} a_{40}}{a_{20} }\sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \phi _{40}\\&+\,\frac{\beta _{4} }{8\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40} }{a_{20} }\sin \phi _{40} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{2} }{8}a_{20} a_{40} \cos \phi _{40} +\frac{\gamma _{3} }{4\omega }a_{40}^{2} \sin 2\phi _{40} \\&+\,\frac{3\gamma _{1} }{8\omega }a_{20} a_{40} \sin \phi _{40}\\&-\,\frac{\eta _{2} }{2\omega }\frac{a_{40} }{a_{20} }\sin \phi _{40} \\ J_{51}= & {} -\frac{\eta _{3} }{2\omega _{1} }\sin \phi _{30} , \quad J_{52} =0, J_{53} =0, \\ J_{54}= & {} 0, \quad J_{55} =-\frac{\mu _{1} }{2}, \quad J_{56} =-\frac{\eta _{3} }{2\omega _{1} }a_{10} \cos \phi _{30} , \\ J_{57}= & {} 0, \quad J_{58} =0\\ J_{61}= & {} -\frac{\beta _{2} }{8}a_{30} \sin \phi _{30} -\frac{3\alpha _{1} }{4\omega }a_{10} \\&-\,\frac{3\beta _{1} }{8\omega }a_{30} \cos \phi _{30} -\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10}^{2} }\cos \phi _{30} \\ \end{aligned}$$
$$\begin{aligned}&+\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10}^{2} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\alpha _{6} \omega }{4}a_{10} -\frac{f\Omega ^{2}}{2\omega a_{10}^{2} }\cos \phi _{10} \\&-\,\frac{\eta _{3} }{2\omega _{1}a_{30} }\cos \phi _{30} \\ J_{62}= & {} -\frac{f\Omega ^{2}}{2\omega a_{10} }\sin \phi _{10}\\ J_{63}= & {} \frac{\alpha _{5} \omega }{4}a_{20} \cos 2\phi _{20} -\frac{\alpha _{2} }{4\omega }a_{20} \cos 2\phi _{20} \\&-\,\frac{\beta _{6} }{4}a_{40} \sin \phi _{40} +\frac{\alpha _{4} }{4}a_{20} \sin 2\phi _{20} \\&-\,\frac{\alpha _{7}}{4}a_{20} \sin 2\phi _{20} \\&-\,\frac{\beta _{6} }{8}a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{7} }{2\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \phi _{30} -\frac{\beta _{4} }{4\omega }a_{40} \cos \phi _{40}\\&-\,\frac{\beta _{7} }{4\omega }\frac{a_{20} a_{30} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} }\right) \\&-\,\frac{\beta _{4} }{8\omega }a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\alpha _{2} }{2\omega }a_{20} -\frac{\omega \alpha _{5} }{2}a_{20} \\ J_{64}= & {} -\frac{\alpha _{5} \omega }{4}a_{20}^{2} \sin 2\phi _{20} +\frac{\alpha _{2} }{4\omega }a_{20}^{2} \sin 2\phi _{20}\\&+\,\frac{\alpha _{4} }{4}a_{20}^{2} \cos 2\phi _{20} -\frac{\alpha _{7} }{4}a_{20}^{2} \cos 2\phi _{20} \\&-\,\frac{\beta _{6} }{4}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40}} \right) \\&+\,\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\ J_{65}= & {} -\frac{\beta _{2} }{8}a_{10} \sin \phi _{30} -\frac{3\beta _{1} }{8\omega }a_{10} \cos \phi _{30} \\&+\,\frac{\eta _{1} }{2\omega a_{10} }\cos \phi _{30} -\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} }{a_{10} }\cos \phi _{30}\\&-\,\frac{\beta _{3} }{4\omega }a_{30} \cos 2\phi _{30} \\&-\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} }{a_{10} }\cos \left( {2\phi _{20} +\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\beta _{3} }{2\omega }a_{30} +\frac{\eta _{3} }{2\omega _{1} }\frac{a_{10} }{a_{30}^{2} }\cos \phi _{30} \\ J_{66}= & {} -\frac{\beta _{2} }{8}a_{10} a_{30} \cos \phi _{30} +\frac{3\beta _{1} }{8\omega }a_{10} a_{30} \sin \phi _{30}\\&-\,\frac{\eta _{1} }{2\omega }\frac{a_{30} }{a_{10} }\sin \phi _{30} +\frac{\beta _{7} }{4\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \phi _{30} \\&+\,\frac{\beta _{3} }{4\omega }a_{30}^{2} \sin 2\phi _{30} \\&+\,\frac{\beta _{7} }{8\omega }\frac{a_{20}^{2} a_{30} }{a_{10} }\sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\eta _{3} }{2\omega _{1} }\frac{a_{10} }{a_{30} }\sin \phi _{30} \\ J_{67}= & {} -\frac{\beta _{6} }{4}a_{20} \sin \phi _{40} \\&-\,\frac{\beta _{6} }{8}a_{20} \sin \left( {2\phi _{20} -\phi _{40} } \right) -\frac{\beta _{4} }{4\omega }a_{20} \cos \phi _{40}\\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\beta _{4} }{8\omega }a_{20} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{5} }{4\omega }a_{40} \cos \left( {2\phi _{20} -2\phi _{40} } \right) -\frac{\beta _{5} }{2\omega }a_{40} \\ J_{68}= & {} -\frac{\beta _{6} }{4}a_{20} a_{40} \cos \phi _{40}\\&+\,\frac{\beta _{6} }{8}a_{20} a_{40} \cos \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\beta _{4} }{4\omega }a_{20} a_{40} \sin \phi _{40}\\&-\,\frac{\beta _{4} }{8\omega }a_{20} a_{40} \sin \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\beta _{5} }{4\omega }a_{40}^{2} \sin \left( {2\phi _{20} -2\phi _{40} } \right) \\ J_{71}= & {} 0, J_{72} =0, \\ J_{73}= & {} -\frac{\eta _{4} }{2\omega _{2} }\sin \phi _{40} , \quad J_{74} =0\\ J_{75}= & {} 0, J_{76} =0, \quad J_{77} =-\frac{\mu _{2} }{2}, \\ \end{aligned}$$
$$\begin{aligned} J_{78}= & {} -\frac{\eta _{4} }{2\omega _{2} }a_{20} \cos \phi _{40} \\ J_{81}= & {} -\frac{\gamma _{6} }{4}a_{30} \sin \phi _{30} -\frac{\alpha _{4} }{4}a_{10} \sin 2\phi _{20} \\&+\,\frac{\alpha _{7} }{4}a_{10} \sin 2\phi _{20} +\frac{\gamma _{6} }{8}a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\gamma _{4} }{4\omega }a_{30} \cos \phi _{30} \\&-\,\frac{\gamma _{7} }{2\omega }\frac{a_{10} a_{40} }{a_{20} }\cos \phi _{40} -\frac{\alpha _{2} }{4\omega }a_{10} \cos 2\phi _{20}\\&+\,\frac{\alpha _{5} \omega }{4}a_{10} \cos 2\phi _{20} -\frac{\gamma _{4} }{8\omega }a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\gamma _{7} }{4\omega }\frac{a_{10} a_{40} }{a_{20} }\cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\alpha _{5} \omega }{2}a_{10} -\frac{\alpha _{2} }{2\omega }a_{10} \\ \end{aligned}$$
$$\begin{aligned} J_{82}= & {} \frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20} }\cos \left( {\phi _{10} +\phi _{20} } \right) \\ J_{83}= & {} -\frac{\gamma _{2} }{8}a_{40} \sin \phi _{40} -\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20}^{2} }\sin \left( {\phi _{10} +\phi _{20} } \right) \\&+\,\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40} }{a_{20}^{2} }\cos \phi _{40} -\frac{3\gamma _{1} }{8\omega }a_{40} \cos \phi _{40}\\&-\,\frac{\eta _{2} }{2\omega }\frac{a_{40} }{a_{20}^{2} }\cos \phi _{40} \\&+\,\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} a_{40} }{a_{20}^{2} }\cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\alpha _{6} \omega }{4}a_{20} -\frac{3\alpha _{1} }{4\omega }a_{20} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\eta _{4}}{2\omega _{2} a_{40} }\cos \phi _{40} \\ J_{84}= & {} -\frac{\alpha _{4} }{4}a_{10}^{2} \cos 2\phi _{20}\\&+\,\frac{\alpha _{7} }{4}a_{10}^{2} \cos 2\phi _{20} \\&+\,\frac{f\Omega ^{2}}{2\omega }\frac{1}{a_{20} }\cos \left( {\phi _{10} +\phi _{20} }\right) \\&+\,\frac{\gamma _{6} }{4}a_{10} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\alpha _{2} }{4\omega }a_{10}^{2} \sin 2\phi _{20}\\&-\,\frac{\alpha _{5} \omega }{4}a_{10}^{2} \sin 2\phi _{20}\\&+\,\frac{\gamma _{4} }{4\omega }a_{10} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{5} }{4\omega }a_{30}^{2} \sin \left( {2\phi _{20} +2\phi _{30} } \right) \\&+\,\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40} }{a_{20} }\sin \left( {2\phi _{20} -\phi _{40} } \right) \\ J_{85}= & {} -\frac{\gamma _{6} }{4}a_{10} \sin \phi _{30} \\&+\,\frac{\gamma _{6} }{8}a_{10} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&-\,\frac{\gamma _{4} }{4\omega }a_{10} \cos \phi _{30}\\&-\,\frac{\gamma _{4} }{8\omega }a_{10} \cos \left( {2\phi _{20} +\phi _{30}}\right) \\&-\,\frac{\gamma _{5} }{4\omega }a_{30} \cos \left( {2\phi _{20} +2\phi _{30} } \right) \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{5} }{2\omega }a_{30} \\ J_{86}= & {} -\frac{\gamma _{6} }{4}a_{10} a_{30} \cos \phi _{30}\\&+\,\frac{\gamma _{6} }{8}a_{10} a_{30} \cos \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{4} }{4\omega }a_{10} a_{30} \sin \phi _{30}\\&+\,\frac{\gamma _{4} }{8\omega }a_{10} a_{30} \sin \left( {2\phi _{20} +\phi _{30} } \right) \\&+\,\frac{\gamma _{5} }{4\omega }a_{30}^{2} \sin \left( {2\phi _{20} +2\phi _{30} } \right) \\ J_{87}= & {} -\frac{\gamma _{2} }{8}a_{20} \sin \phi _{40} -\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} }{a_{20} }\cos \phi _{40}\\&-\,\frac{3\gamma _{1} }{8\omega }a_{20} \cos \phi _{40} +\frac{\eta _{2} }{2\omega a_{20} }\cos \phi _{40} \\&-\,\frac{\gamma _{3} }{4\omega }a_{40} \cos 2\phi _{40} \\ \end{aligned}$$
$$\begin{aligned}&-\,\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} }{a_{20} }\cos \left( {2\phi _{20} -\phi _{40} } \right) \\&-\,\frac{\gamma _{3} }{2\omega }a_{40} +\frac{\eta _{4} }{2\omega _{2} }\frac{a_{20}}{a_{40}^{2} }\cos \phi _{40} \\ J_{88}= & {} -\frac{\gamma _{2} }{8}a_{20} a_{40} \cos \phi _{40} +\frac{\gamma _{7} }{4\omega }\frac{a_{10}^{2} a_{40} }{a_{20} }\sin \phi _{40} \\&+\,\frac{3\gamma _{1} }{8\omega }a_{20} a_{40} \sin \phi _{40} -\frac{\eta _{2} }{2\omega }\frac{a_{40} }{a_{20} }\sin \phi _{40} \\&+\,\frac{\gamma _{3} }{4\omega }a_{40}^{2} \sin 2\phi _{40} -\frac{\gamma _{7} }{8\omega }\frac{a_{10}^{2} a_{40} }{a_{20} }\sin \left( {2\phi _{20} -\phi _{40} } \right) \\&+\,\frac{\eta _{4} }{2\omega _{2} }\frac{a_{20} }{a_{40} }\sin \phi _{40} \end{aligned}$$
