Nonlinear coupling of transverse modes of a fixed–fixed microbeam under direct and parametric excitation

Abstract

Tuning of linear frequency and nonlinear frequency response of microelectromechanical systems is important in order to obtain high operating bandwidth. Linear frequency tuning can be achieved through various mechanisms such as heating and softening due to DC voltage. Nonlinear frequency response is influenced by nonlinear stiffness, quality factor and forcing. In this paper, we present the influence of nonlinear coupling in tuning the nonlinear frequency response of two transverse modes of a fixed–fixed microbeam under the influence of direct and parametric forces near and below the coupling regions. To do the analysis, we use nonlinear equation governing the motion along in-plane and out-of-plane directions. For a given DC and AC forcing, we obtain static and dynamic equations using the Galerkin’s method based on first-mode approximation under the two different resonant conditions. First, we consider one-to-one internal resonance condition in which the linear frequencies of two transverse modes show coupling. Second, we consider the case in which the linear frequencies of two transverse modes are uncoupled. To obtain the nonlinear frequency response under both the conditions, we solve the dynamic equation with the method of multiple scale (MMS). After validating the results obtained using MMS with the numerical simulation of modal equation, we discuss the influence of linear and nonlinear coupling on the frequency response of the in-plane and out-of-plane motion of fixed–fixed beam. We also analyzed the influence of quality factor on the frequency response of the beams near the coupling region. We found that the nonlinear response shows single curve near the coupling region with wider width for low value of quality factor, and it shows two different curves when the quality factor is high. Consequently, we can effectively tune the quality factor and forcing to obtain different types of coupled response of two modes of a fixed–fixed microbeam.

Appendices

Appendix 1

Nonlinear static equations

$$\begin{aligned}&758.48\,{q_{{1}}}^{7}\alpha _{{1}}+ \left( 1022.84\,r_{{1}} \alpha _{{1}}- 1022.84\,\alpha _{{1}} \right) {q_{{1}}}^{6}\\&\quad +\, \big ( 347.07\,\alpha _{{1}}+ 347.07\,{r_{{1}}}^{2} \alpha _{{1}}+ 758.48\alpha _{{2}}{q_{{2}}}^{2}\\&\quad +\, 61.66\,N+\, 1934.02- 1388.29\,r_{{1}}\alpha _{{1}} \big ) {q_{{1}}}^{5}\\&\quad +\, \big ( 2654.13\,r_{{1}}- 470.92 {r_{{1}}}^{2}\alpha _{{1}}\\&\quad -\, 2654.13- 1022.84\alpha _{{2}}{q_{{2}}}^{2}+ 83.15\,r_{ {1}}N\\&\quad +\, 1022.84\, r_{{1}}\alpha _{{2}}{q_{{2}}}^{2}+ 470.92\,r_{ {1}}\alpha _{{1}} \\&\quad -\, 83.15\,N \big ) {q_{{1}}}^{4}+ \big ( 347.07{r_{{1}}}^{2}\alpha _{{2}}{q_{{2}}}^{2} + 927.00\\&\quad +\, 28.21\,N- 3708.01\,r_{{1}}+ 927.00\,{r_{{1}}}^{2} \\&\quad +\, 151.34\,{r_{{1}}}^{2}\alpha _{{1}}- 112.85\,r_{{1}}N- 1388.29r_{{1}}\alpha _{{2}}{q_{{2}}}^{2}\\&\quad +\, 347.07\,\alpha _{{ 2}}{q_{{2}}}^{2}+ 28.21\,{r_{{1}}}^{2}N \big ) \\&\quad \times \,{q_{{1}}}^{3} + \big ( 38.28\,r_{{1}}N- 1.33\,\beta _{{s}}{V_{{10}}}^{2} \\&\quad +\, 1330.88\,r_{{1}}- 470.92\,{r_{{1}}}^{2}\alpha _{{2}}{q_{{2}} }^{2}+ 470.92\,r_{{1}}\alpha _{{2}}{q_{{2}}}^{2} \\&\quad -\, 1330.88\,{r_ {{1}}}^{2}- 38.28\,{r_{{1}}}^{2}N+ 1.33\,\beta _{{s}}{V_{{ 12}}}^{2} \big ) \\&\quad \times \,{q_{{1}}}^{2}+ \big ( 12.30\,{r_{{1}}}^{2}N- 2.00\,\beta _{{s}}{V_{{12}}}^{2} \\&\quad -\, 2.00\,\beta _{{s}}{V_{{ 10}}}^{2}r_{{1}}+ 500.56\,{r_{{1}}}^{2}+ 151.34\,{r_{{1}}}^{ 2}\alpha _{{2}}{q_{{2}}}^{2} \big ) \\&\quad \times q_{{1}}+ 0.83\,\beta _{{s}} {V_{{12}}}^{2} - 0.83\,\beta _{{s}}{V_{{10}}}^{2}{r_{{1}}}^{2}=0 \end{aligned}$$

$$\begin{aligned}&347.07\,{q_{{2}}}^{5}\alpha _{{3}}\alpha _{{2}}- 470.92\,{q_{{2}}}^{4} \alpha _{{3}}\alpha _{{2}}\\&\quad +\, \big ( 151.34\,\alpha _{{3}}\alpha _{{2}}- 1.85\,{V_{{g}}}^{2}\beta _{{3\,g}}\\&\quad +\, 927.0+ 347.07 \alpha _{{3}}\alpha _ {{1}}{q_{{1}}}^{2}+ 1.85\,{V_{{10}}}^{2}\alpha _{{2\,g}}\\&\quad +\, 1.85\,{V_{{12 }}}^{2}\alpha _{{2\,g}}+ 28.21\,\alpha _{{3}}N \big ) {q_{{2}}}^{3}+ \big (-\!2.66 {V_{{12}}}^{2}\alpha _{{2\,g}}\\&\quad -\, 38.28\,\alpha _{{3}}N- 2.66\,{V_{{10}}}^{2}\alpha _{{2\,g}}- 1330.88\\&\quad +\, 1.33\,{V_{{10}}}^{2} \alpha _{{g}}+ 1.33\,{V_{{12}}}^{2}\alpha _{{g}} + 3.99{V_{{g}}}^{2} \beta _{{3\,g}}\\&\quad -\, 1.33\,{V_{{g}}}^{2}\beta _{{2\,g}}- 470.92\,\alpha _{{3} }\alpha _{{1}}{q_{{1}}}^{2} \big ) {q_{{2}}}^{2}\\&\quad +\, \big ( - 2.0\,{V_{{ 12}}}^{2}\alpha _{{g}}+ 500.56 + 151.34\alpha _{{3}}\alpha _{{1}}{q_{{1} }}^{2}\\&\quad -\, 3.0\,{V_{{g}}}^{2}\beta _{{3\,g}}+ 2.0\,{V_{{g}}}^{2}\beta _{{2 \,g}}+ 12.30\alpha _{{3}}N\\&\quad +\, 1.0\,{V_{{12}}}^{2}\alpha _{{2\,g}} + 1.0\, {V_{{10}}}^{2}\alpha _{{2\,g}}- 2.0\,{V_{{10}}}^{2}\alpha _{{g}} \big )\\&\quad \times \, q_{{2}}- 0.831\,{V_{{g}}}^{2}\beta _{{g}}- 0.831\,{V_{{g}}}^{2 }\beta _{{2\,g}} \\&\quad +\, 0.831\,{V_{{g}}}^{2}\beta _{{3\,g}}+ 0.831\,{V_{{10}}} ^{2}\alpha _{{g}}\\&\quad +\, 0.831\,{V_{{12}}}^{2}\alpha _{{g}}=0 \end{aligned}$$

In-plane equation coefficients

$$\begin{aligned} m_{1i}= & {} 5.56\,{{ q_{1}}}^{2}{r_{{1}}}^{3}- 11.59\,{{ q_{1}}}^{4} {r_{{1}}}^{2}- 17.11\,{{ q_{1}}}^{5}r_{{1}}\\&+\, 3.98\,{r_{{1 }}}^{2}{ q_{1}}- 2.65\,{{ q_{1}}}^{3}{r_{{1}}}^{3} + 23.86 \,{r_{{1}}}^{2}{{ q_{1}}}^{3}\\&-\, 3.98\,{ q_{1}}\,{r_{{1}}}^{3}+ 34.77\,{{ q_{1}}}^{4}r_{{1}}- 16.67\,{{ q_{1}}}^{2}{r_{{1 }}}^{2}\\&-\, 23.86\,{{ q_{1}}}^{3}r_{{1}} + 5.55\,r_{{1}}{{ q_{1}}}^{2}+ {r_{{1}}}^{3}+ 2.65\,{{ q_{1}}}^{3}\\&-\, 8.49\,{{ q_{1}}}^{6}- 11.59\,{{ q_{1}}}^{4}+ 17.11 \,{{ q_{1}}}^{5} \end{aligned}$$

$$\begin{aligned} k_{1i}= & {} 511.42\,{{ q_{1}}}^{3}{r_{{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2}+ 2119.14\,{r_{{1}}}^{2}{{ q_{1}}}^{3}\alpha _{{1}} \\&-\, 1534.26{{ q_{1}}} ^{5}{r_{{1}}}^{3}\alpha _{{1}}- 11.11\beta _{{s}}{V_{{12}}}^{2}{{ q_{1}}}^{2} \\&+\, 151.34\,{r_{{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2}+ 276.20\,{{ q_{1}}}^{5}N \\&-\, 6826.314\,{{ q_{1}}}^{6}{r_{{1}}}^{2}\alpha _{{1}} + 13808.31\,{{ q_{1}}}^{5}{r_{{1}}}^{2}\alpha _{{1}} \\&- \,41.57\,{{ q_{1}}}^ {3}{r_{{1}}}^{3}N-2\,\beta _{{s}}{V_{{12}}}^{2} \\&-\, 11.11\,\beta _{{s}}{V_{ {10}}}^{2}r_{{1}}{{ q_{1}}}^{2}\\&+\, 554.90\,{{ q_{1}}}^{4}r_{{1}}N 41.57 \,{{ q_{1}}}^{3}N \\&+\, 374.15\,{r_{{1}}}^{2}{{ q_{1}}}^{3}N+500.56\,{r_{{ 1}}}^{3} \\&-\, 5802.07\,{{ q_{1}}}^{4}+ 8563.10\,{{ q_{1}}}^{5} 253.92\,{{ q_{1}}}^{2}{r_{{1}}}^{2}N \\&+\, 1327.064733\,{{ q_{1}}}^{3} - 4252.39{{ q_{1}}}^{6}\\&+ 2781.01{{ q_{1}}}^{2} {r_{{1}}}^{3} \\&+\, 57.42{r_{{1}}}^{ 2}{ q_{1}}\,N+ 511.42\,{{ q_{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2} \\&-\, 5802.07\,{{ q_{1}}}^{4}{r_{{1}}}^{2}- 8563.10\,{{ q_{1}}}^{5}r_{{1}} \\&+\, 1996.33{r_{{1}}}^{2}{ q_{1}} -1327.06\,{{ q_{1}}}^{3}{r_{{1}}}^{3}\\&+\, 11943.58\,{r_{{1}}}^{2}{{ q_{1}}}^{3} - 1996.33\,{ q_{1}}\,{r_{{1}}}^{ 3}\\&+\, 17406.21\,{{ q_{1}}}^{4}r_{{1}} 8343.02\,{{ q_{1}}}^{2}{r_{{1}}}^ {2} \\&-\, 11943.58\,{{ q_{1}}}^{3}r_{{1}}+ 2781.01\,r_{{1}}{{ q_{1}}}^{2} \\&+\, 3397.83\,{{ q_{1}}}^{5}\alpha _{{2}}{{ q_{2}}}^{2} - 3123.66\,{{ q_{1}} }^{2}{r_{{1}}}^{2}\alpha _{{2}}{{ q_{2}}}^{2} \\&+\, 6826.31\,{{ q_{1}}}^{4}r _{{1}}\alpha _{{2}}{{ q_{2}}}^{2}- 184.97\,{{ q_{1}}}^{4}N \\&-\, 138.32\,{{ q_{1}}}^{6}N 12.31\,{r_{{1}}}^{3}N - 6826.31\,{{ q_{1}}}^{6}\alpha _{{ 1}}\\&+\, 10193.49\,{{ q_{1}}}^{7}\alpha _{{1}} - 5104.69\,{{ q_{1}}}^{8} \alpha _{{1}}\\&+\, 1534.26{{ q_{1}}}^{5}\alpha _{{1}}+ 454.01{{ q_{1}}}^ {2}{r_{{1}}}^{3}\alpha _{{1}} \\&-\,13808.31{{ q_{1}}}^{5}r_{{1}}\alpha _{{ 1}} -10193.49{{ q_{1}}}^{7}r_{{1}}\alpha _{{1}}\\&+\,3123.66{{ q_{1}}}^ {4}{r_{{1}}}^{3}\alpha _{{1}} + 7.98\,\beta _{{s}}{V_{{12}}}^{2}{ q_{1}}\\&+\, 84.64\,{{ q_{1}}}^{2}{r_{{1}}}^{3}N + 3123.66{{ q_{1}}}^{4}r_{{1}} \alpha _{{1}} \\&-\, 276.21\,{{ q_{1}}}^{5}r_{{1}}N -2119.14\,{{ q_{1}}}^{3}{ r_{{1}}}^{3}\alpha _{{1}} \\&+\, 5.30\,\beta _{{s}}{V_{{12}}}^{2}{{ q_{1}}}^{3 }+ 20478.94\,{{ q_{1}}}^{6}r_{{1}}\alpha _{{1}} \\&-\, 5.30\,\beta _{{s}}{V_{{ 10}}}^{2}{{ q_{1}}}^{3}- 374.15\,{{ q_{1}}}^{3}r_{{1}}N \\&-\,57.42\,{ q_{1}}\,{r_{{1}}}^{3}N-2\,\beta _{{s}}{V_{{10}}}^{2}{r_{{1}}}^{3} \\&-\, 9370.99 \,{{ q_{1}}}^{4}{r_{{1}}}^{2} \alpha _{{1}}- 2275.44\,{{ q_{1}}}^{4} \alpha _{{2}}{{ q_{2}}}^{2} \\&-\, 1701.56\,{{ q_{1}}}^{6}\alpha _{{2}}{{ q_{2}}}^{2} + 84.64\,r_{{1}}{{ q_{1}}}^{2}N \\&+\, 706.38\,{r_{{1}}}^{2}{ q_{1}} \,\alpha _{{2}}{{ q_{2}}}^{2} - 7.97\,\beta _{{s}}{V_{{10}}}^{2}{r_{{1}}} ^{2}{ q_{1}}\\&- \,706.38\,{ q_{1}}\,{r_{{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2 } - 4602.77\,{{ q_{1}}}^{3} r_{{1}}\alpha _{{2}}{{ q_{2}}}^{2}\\&+\,1041.22\, {{ q_{1}}}^{2}{r_{{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2} + 4602.77\,{r_{{1 }}}^{2}{{ q_{1}}}^{3}\alpha _{{2}}{{ q_{2}}}^{2} \\&-\, 3397.83\,{{ q_{1}}}^{ 5}r_{{1}}\alpha _{{2}}{{ q_{2}}}^{2} 184.97\,{{ q_{1}}}^{4}{r_{{1}}}^{ 2}N \\&-\, 2275.44\,{{ q_{1}}}^{4}{r_{{1}}}^{2}\alpha _{{2}}{{ q_{2}}}^{2} + 1041.22\,r_{{1}}{{ q_{1}}}^{2}\alpha _{{2}}{{ q_{2}}}^{2} \end{aligned}$$

$$\begin{aligned} n_{1i}= & {} 1041.22\,{{ q_{1}}}^{2}{r_{{1}}}^{3}\alpha _{{1}} + 511.42\,{{ q_{1}}}^ {3}\alpha _{{1}}\\&- 706.38\,{ q_{1}}\,{r_{{1}}}^{3}\alpha _{{1}}s+ 151.33\, {r_{{1}}}^{3}\alpha _{{1}}\\&-\, 3123.66\,{r_{{1}}}^{2}{{ q_{1}}}^{2}\alpha _ {{1}}+706.38\,{r_{{1}}}^{2}{ q_{1}}\,\alpha _{{1}}\\&+\, 6826.31\,{{ q_{1}} }^{4}r_{{1}}\alpha _{{1}}- 2275.44\,{{ q_{1}}}^{4}\alpha _{{1}} \\&-\, 511.42 \,{{ q_{1}}}^{3}{r_{{1}}}^{3}\alpha _{{1}}- 2275.44\,{{ q_{1}}}^{4}{r_{ {1}}}^{2}\alpha _{{1}}\\&-\,1701.56\,{{ q_{1}}}^{6}\alpha _{{1}}+ 1041.22\,r _{{1}}{{ q_{1}}}^{2}\alpha _{{1}} \\&+\, 74602.77{r_{{1}}}^{2}{{ q_{1}}}^{3} \alpha _{{1}}- 3397.83{{ q_{1}}}^{5}r_{{1}}\alpha _{{1}}\\&+\, 3397.83{{ q_{1}}}^{5}\alpha _{{1}} - 4602.77{{ q_{1}}}^{3}r_{{1}}\alpha _{{1}} \\ n_{2i}= & {} - 6826.31{{ q_{1}}}^{5}{r_{{1}}}^{2}\alpha _{{1}}- 5104.69{{ q_{1}} }^{7}\alpha _{{1}}\\&-\,13808.31{{ q_{1}}}^{4}r_{{1}}\alpha _{{1}}- 9370.99{r_{{1}}}^{2}{{ q_{1}}}^{3}\alpha _{{1}} \\&+\, 1534.26\,{{ q_{1}}} ^{4}\alpha _{{1}}+ 10193.49\,{{ q_{1}}}^{6}\alpha _{{1}}\\&+\, 13808.31\,{{ q_{1}}}^{4}{r_{{1}}}^{2}\alpha _{{1}}+ 454.01\,{ q_{1}}\,{r_{{1}}}^{3} \alpha _{{1}} \\&- \, 1534.26\,{{ q_{1}}}^{4}{r_{{1}}}^{3}\alpha _{{1}}+ 3123.66\,{{ q_{1}}}^{3}{r_{{1}}}^{3}\alpha _{{1}}\\&-\, 6826.31\,{{ q_{1}}} ^{5}\alpha _{{1}}- 2119.14\,{{ q_{1}}}^{2}{r_{{1}}}^{3}\alpha _{{1}}\\&+\, 3123.66\,{{ q_{1}}}^{3}r_{{1}}\alpha _{{1}}+ 20478.94\,{{ q_{1}}}^{5}r _{{1}}\alpha _{{1}} \\&-\,10193.49\,{{ q_{1}}}^{6}r_{{1}}\alpha _{{1}} + 2119.14\,{r_{{1}}}^{2}{{ q_{1}}}^{2}\alpha _{{1}}\\ n_{3i}= & {} - 4602.77\,{{ q_{1}}}^{3}r_{{1}}\alpha _{{2}}- 2275.44\,{{ q_{1}}}^{4}{ r_{{1}}}^{2}\alpha _{{2}}\\&-\, 3123.66\,{{ q_{1}}}^{2}{r_{{1}}}^{2}\alpha _{ {2}}+ 4602.77\,{r_{{1}}}^{2} {{ q_{1}}}^{3}\alpha _{{2}} \\&-\,3397.83{{ q_{1}}}^{5}r_{{1}}\alpha _{{2}}- 1701.56{{ q_{1}}}^{6}\alpha _{{2}}\\&+\, 706.38{r_{{1}}}^{2}{ q_{1}}\alpha _{{2}}+ 511.42{{ q_{1}}}^{3} \alpha _{{2}}+ 151.34\,{r_{{1}}}^{3}\alpha _{{2}}\\&+\,6826.31{{ q_{1}}}^{ 4}r_{{1}}\alpha _{{2}}+ 1041.22{{ q_{1}}}^{2}{r_{{1}}}^{3}\alpha _{{2} }\\&+\, 1041.22r_{{1}}{{ q_{1}}}^{2} \alpha _{{2}}- 511.42{{ q_{1}}}^{3}{ r_{{1}}}^{3}\alpha _{{2}}\\&- 706.38{ q_{1}}{r_{{1}}}^{3}\alpha _{{2}}{-} 2275.44{{ q_{1}}}^{4}\alpha _{{2}}{+} 3397.83{{ q_{1}}}^{5}\alpha _{{ 2}}\\ n_{4i}= & {} - 4550.87{{ q_{1}}}^{4}\alpha _{{2}}{ q_{2}}- 1022.84{{ q_{1}}}^{3} {r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}\\&-\, 4550.87{{ q_{1}}}^{4}{r_{{1}}}^{2 }\alpha _{{2}}{ q_{2}}\\&+\,13652.63 {{ q_{1}}}^{4}r_{{1}}\alpha _{{2}}{ q_{2}}+9205.54\,{{ q_{1}}}^{3}{r_{{1}}}^{2}\alpha _{{2}}{ q_{2}}\\&-\, 6795.66\,{{ q_{1}}}^{5}r_{{1}}\alpha _{{2}}{ q_{2}}- 9205.54\,{{ q_{1} }}^{3}r_{{1}}\alpha _{{2}}{ q_{2}}\\&+\, 1412.76\,{r_{{1}}}^{2}{ q_{1}}\, \alpha _{{2}}{ q_{2}}-1412.76\\&\times \,{ q_{1}}\,{r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}- 6247.33\,{r_{{1}}}^{2}{{ q_{1}}}^{2}\alpha _{{2}}{ q_{2}}\\&+\, 2082.44\,r_{{1}}{{ q_{1}}}^{2}\alpha _{{2}}{ q_{2}}+2082.44\,{{ q_{1} }}^{2}{r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}\\&+\,1022.84\,{{ q_{1}}}^{3}\alpha _{{2}}{ q_{2}}+ 302.67 {r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}\\&+\, 6795.66\,{{ q_{1}}}^{5}\alpha _{{2}}{ q_{2}}- 3403.13\,{{ q_{1}}}^{6}\alpha _{{2}}{ q_{2}}\\ n_{5i}= & {} - 2.0\,\beta _{{s}}{r_{{1}}}^{3}- 7.98\,\beta _{{s}}{r_{{1}}}^{2}{ q_{1} }\\&-\, 11.11\,\beta _{{s}}r_{{1}}{{ q_{1}}}^{2}- 5.30\,\beta _{{s}}{{ q_{1}} }^{3}\\ n_{6i}= & {} 3.70\,\beta _{{s}}{{ q_{1}}}^{3}- 1.66\,\beta _{{s}}+ 6.0\,\beta _{{s}}{ q_{1}}- 7.98\,\beta _{{s}}{{ q_{1}}}^{2} \end{aligned}$$

$$\begin{aligned} n_{7i}= & {} 1041.22{{ q_{1}}}^{3}r_{{1}}\alpha _{{2}}- 706.38{{ q_{1}}}^{2}{r_ {{1}}}^{3}\alpha _{{2}}\\&-\, 2275.44{{ q_{1}}}^{5}{r_{{1}}}^{2}\alpha _{{2 }}+ 151.33\,{ q_{1}}{r_{{1}}}^{3}\alpha _{{2}}\\&-\, 3397.83\,{{ q_{1}}}^{ 6}r_{{1}}\alpha _{{2}}-\,4602.77\,{{ q_{1}}}^{4}r_{{1}}\alpha _{{2}}\\&+\,3397.83\,{{ q_{1}}}^{6}\alpha _{{2}}+\, 511.42\,{{ q_{1}}}^{4}\alpha _{{2 }}\\&+\, 1041.22\,{{ q_{1}}}^{3}{r_{{1}}}^{3}\alpha _{{2}}-\, 1701.56\,{{ q_{1}}}^{7}\alpha _{{2}}\\&-\,3123.66\,{r_{{1}}}^{2}{{ q_{1}}}^{3}\alpha _{{2}} + 4602.77\,{{ q_{1}}}^{4}{r_{{1}}}^{2} \alpha _{{2}}\\&-\,511.42{{ q_{1}}} ^{4}{r_{{1}}}^{3}\alpha _{{2}}-\, 2275.44{{ q_{1}}}^{5}\alpha _{{2}}\\&+\, 6826.31{{ q_{1}}}^{5}r_{{1}}\alpha _{{2}}+\, 706.38{{ q_{1}}}^{2}{r_ {{1}}}^{2}\alpha _{{2}}\\ n_{8i}= & {} - 6795.66{{ q_{1}}}^{6}r_{{1}}\alpha _{{2}}{ q_{2}}- 9205.54{{ q_{1}}}^{4}r_{{1}}\alpha _{{2}}{ q_{2}}\\&-\, 1412.76{{ q_{1}}}^{2}{r_{{1}}}^ {3}\alpha _{{2}}{ q_{2}}-\, 6247.33{{ q_{1}}}^{3} {r_{{1}}}^{2}\alpha _{{ 2}}{ q_{2}}\\&-\,4550.87{{ q_{1}}}^{5}{r_{{1}}}^{2}\alpha _{{2}}{ q_{2}}+ 2082.44{{ q_{1}}}^{3}{r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}\\&+\, 1412.76{r _{{1}}}^{2}{{ q_{1}}}^{2}\alpha _{{2}}{ q_{2}} + 302.67\,{ q_{1}}\,{r_{{ 1}}}^{3}\alpha _{{2}}{ q_{2}}\\&+\,1022.84\,{{ q_{1}}}^{4}\alpha _{{2}}{ q_{2}}- 1022.84\,{{ q_{1}}}^{4}{r_{{1}}}^{3}\alpha _{{2}}{ q_{2}}\\&+\, 6795.66 \,{{ q_{1}}}^{6} \alpha _{{2}}{ q_{2}}-\,4550.87\,{{ q_{1}}}^{5}\alpha _{{ 2}}{ q_{2}}\\&+\,2082.44\,{{ q_{1}}}^{3}r_{{1}}\alpha _{{2}}{ q_{2}}+\, 13652.63\,{{ q_{1}}}^{5}r_{{1}}\alpha _{{2}}{ q_{2}}\\&- \, 3403.13\,{{ q_{1}}}^{7}\alpha _{{2}}{ q_{2}}+\, 9205.54\,{{ q_{1}}}^{4}{r_{{1}}}^{2} \alpha _{{2}}{ q_{2}} \end{aligned}$$

$$\begin{aligned} n_{9i}= & {} - 3.98\,\beta _{{s}}r_{{1}}{{ q_{1}}}^{2}+ 2.65\,\beta _{{s}}{{ q_{1}}}^ {4}\\&+\, 3.98\,\beta _{{s}}{r_{{1}}}^{2}{{ q_{1}}}^{2}+ 5.55\,\beta _{{s}}r_ {{1}}{{ q_{1}}}^{3}\\&+\,\beta _{{s}}{r_{{1}}}^{3}{ q_{1}} - 1.85\,\beta _{{s} }{{ q_{1}}}^{3}-0.83\,\beta _{{s}}{r_{{1}}}^{3}\\&-\, 3.0\,\beta _{{s}}{r_{{ 1}}}^{2}{ q_{1}}\\ n_{10i}= & {} - 3.0\,\beta _{{s}}{{ q_{1}}}^{2}- 2.49\,\beta _{{s}}r_{{1}}{ q_{1}}\\&-\, 1.33\,\beta _{{s}}r_{{1}}{{ q_{1}}}^{3}+\beta _{{s}}r_{{1}}- 1.85\, \beta _{{s}}{{ q_{1}}}^{4}\\&+\, 3.98\,\beta _{{s}}{{ q_{1}}}^{3} + 0.83\, \beta _{{s}}{ q_{1}}\\&+\,3.0\,\beta _{{s}}r_{{1}}{{ q_{1}}}^{2}\\ n_{11i}= & {} - 8.49\,{{ q_{1}}}^{6}c_{{1}}- 11.59\,{{ q_{1}}}^{4}{r_{{1}}}^{2}c_{{1 }}\\&+\, 3.98\,{r_{{1}}}^{2}{ q_{1}}\,c_{{1}}+ 5.55\,{{ q_{1}}}^{2}{r_{{1}} }^{3}c_{{1}}\\&+\, 5.55\,r_{{1}}{{ q_{1}}}^{2}c_{{1}} - 3.98\,{ q_{1}}\,{r_{ {1}}}^{3}c_{{1}}-23.86\,{{ q_{1}}}^{3}r_{{1}}c_{{1}}\\&+\, 17.11\,{{ q_{1} }}^{5}c_{{1}}+ 23.86\,{{ q_{1}}}^{3}{r_{{1}}}^{2}c_{{1}}+ 2.65\,{{ q_{1}}}^{3}c_{{1}} \\&-\, 16.67\,{r_{{1}}}^{2}{{ q_{1}}}^{2}c_{{1}}- 11.59\,{{ q_{1}}}^{4}c_{{1}}\\&-\,17.11\,{{ q_{1}}}^{5}r_{{1}}c_{{1}}- 2.65\,{{ q_{1}}}^{3}{r_{{1}}}^{3}c_{{1}}\\&+\,{r_{{1}}}^{3}c_{{1}} + 34.77\,{{ q_{1}}}^{ 4}r_{{1}}c_{{1}} \end{aligned}$$

$$\begin{aligned} \lambda _{1}= & {} \sqrt{\frac{k_{1i}}{m_{1i}}},\quad t_{1}=\frac{n_{1i}}{m_{1i}},\quad t_{2}=\frac{n_{2i}}{m_{1i}}, \\ t_{3}= & {} \frac{n_{3i}}{m_{1i}},\quad t_{4}=\frac{n_{4i}}{m_{1i}},\quad t_{5}=\frac{n_{5i}}{m_{1i}}, \\ t_{6}= & {} \frac{n_{6i}}{m_{1i}},\quad t_{7}=\frac{n_{7i}}{m_{1i}},\quad t_{8}=\frac{n_{8i}}{m_{1i}},\quad t_{9}=\frac{n_{9i}}{m_{1i}}, \\ t_{10}= & {} \frac{n_{10i}}{m_{1i}},\quad t_{11}=\frac{n_{11i}}{m_{1i}} \end{aligned}$$

Out-of-plane equation coefficients

$$\begin{aligned} m_{o}= & {} - 2.65\,{{ q_{2}}}^{3}\alpha _{{3}}+\alpha _{{3}}+ 5.56\,\alpha _{{3}}{{ q_{2}}}^{2}-\, 3.98\,{ q_{2}}\,\alpha _{{3}}\\ k_{o}= & {} - 1996.33\,{ q_{2}}- 1327.06\,{{ q_{2}}}^{3}+ 2781.01\,{{ q_{2}}}^{2}\\&-\, 1534.26\,{{ q_{2}}}^{5}\alpha _{{3}}\alpha _{{2}}-\, 2.65\,\alpha _{{2\,g} }{{ q_{2}}}^{3} {V_{{12}}}^{2}\\&-\,41.57\,{{ q_{2}}}^{3}\alpha _{{3}}N+\,3.98\,{V_{{1g}}}^{2}\beta _{{3\,g}}{ q_{2}}\\&+\, 5.56\,\alpha _{{2\,g}}{{q_{2}}}^{2}{V_{{12}}}^{2}+\, 5.56\,\alpha _{{2\,g}}{{ q_{2}}}^{2} {V_{{10 }}}^{2}\\&-\, 3.98\,\alpha _{{2\,g}}{ q_{2}}\,{V_{{10}}}^{2}+\,84.64\,\alpha _ {{3}}{{ q_{2}}}^{2}N\\&+\, 3123.66\,{{ q_{2}}}^{4}\alpha _{{3}}\alpha _{{2}}+\, 151.34\,\alpha _{{3}} \alpha _{{1}}{{ q_{1}}}^{2}\\&-\, 5.56\,{V_{{1g}}}^{2} \beta _{{3\,g}}{{ q_{2}}}^{2}-\, 57.42\,{ q_{2}}\,\alpha _{{3}}N\\&-\,3.98\, \alpha _{{2\,g}}{ q_{2}}\,{V_{{12}}}^{2}-\, 2119.14 {{ q_{2}}}^{3}\alpha _{{3}}\alpha _{{2}}\\&+\, 454.01\,\alpha _{{3}}{{ q_{2}}}^{2}\alpha _{{2}}+ 500.56 \\&-\, 1.0\,{V_{{1g}}}^{2}\beta _{{3\,g}}-511.42\,{{ q_{2}}}^{3} \alpha _{{3}}\alpha _{{1}}{{ q_{1}}}^{2} \\&+\, 1041.22\,\alpha _{{3}}{{ q_{2}} }^{2}\alpha _{{1}}{{ q_{1}}}^{2}- 706.38\,{ q_{2}}\,\alpha _{{3}}\alpha _ {{1}}{{ q_{1}}}^{2}\\&+\, 1.0\,\alpha _{{2\,g}}{V_{{12}}}^{2}-\,2.65\,\alpha _ {{2\,g}}{{ q_{2}}}^{3} {V_{{10}}}^{2}\\&+\, 2.65\,{V_{{1g}}}^{2}\beta _{{3\,g} }{{ q_{2}}}^{3}+\, 1.0\,\alpha _{{2\,g}}{V_{{10}}}^{2}\\&- \,2.0\,{V_{{1g}}}^{2 }\beta _{{g}}+ 12.30\,\alpha _{{3}}N\\ n_{1o}= & {} - 706.38\,\alpha _{{3}}{ q_{2}}\,\alpha _{{2}}+ 1041.22\,\alpha _{{3}}{{ q_{2}}}^{2}\alpha _{{2}}\\&+\, 151.34\,\alpha _{{3}}\alpha _{{2}}- 511.42\,{{ q_{2}}}^{3}\alpha _{{3}}\alpha _{{2}}\\ n_{2o}= & {} - 1534.26\,{{ q_{2}}}^{4}\alpha _{{3}}\alpha _{{2}}- 2119.14\,\alpha _{{3 }}{{ q_{2}}}^{2}\alpha _{{2}}\\&+\, 3123.66\,{{ q_{2}}}^{3}\alpha _{{3}} \alpha _{{2}} + 454.01\,\alpha _{{3}}{ q_{2}}\,\alpha _{{2}}\\ n_{3o}= & {} 151.34\,\alpha _{{3}}\alpha _{{1}}- 511.42\,{{ q_{2}}}^{3}\alpha _{{3}} \alpha _{{1}}\\&-\, 706.38\,{ q_{2}}\,\alpha _{{3}}\alpha _{{1}}+ 1041.22\, \alpha _{{3}}{{ q_{2}}}^{2}\alpha _{{1}}\\ n_{4o}= & {} - 1412.76\,\alpha _{{3}}{ q_{2}}\,\alpha _{{1}}{ q_{1}}- 1022.84\,{{ q_{2}}}^{3}\alpha _{{3}}\alpha _{{1}}{ q_{1}}\\&+\, 302.67\,\alpha _{{3}}\alpha _{ {1}}{ q_{1}} + 2082.44\,{{ q_{2}}}^{2}\alpha _{{3}}\alpha _{{1}}{ q_{1}}\\ n_{5o}= & {} - 2.0\,\beta _{{g}}- 1.0\,\beta _{{3\,g}}+ 3.98\,\beta _{{3\,g}}{ q_{2}}\\&+\, 2.65\,\beta _{{3\,g}}{{ q_{2}}}^{3}- 5.56\,\beta _{{3\,g}}{{ q_{2}}}^{2 }\\ n_{6o}= & {} 5.56\,\alpha _{{2\,g}}{{ q_{2}}}^{2}- 2.55\,\alpha _{{2\,g}}{{ q_{2}}}^ {3}\\&-\, 3.98\,\alpha _{{2\,g}}{ q_{2}}+ 1.0\,\alpha _{{2\,g}} \end{aligned}$$

$$\begin{aligned} n_{7o}= & {} 151.34\,{ q_{2}}\,\alpha _{{3}}\alpha _{{1}}- 706.38\,\alpha _{{3}}{{ q_{2}}}^{2}\alpha _{{1}}\\&+\, 1041.22\,{{ q_{2}}}^{3}\alpha _{{3}}\alpha _{{ 1}}- 511.42\,{{ q_{2}}}^{4}\alpha _{{3}}\alpha _{{1}}\\ n_{8o}= & {} - 1412.76\,{{ q_{2}}}^{2}\alpha _{{3}}\alpha _{{1}}{ q_{1}}+ 302.67\, \alpha _{{3}}{ q_{2}}\,\alpha _{{1}}{ q_{1}}\\&+\, 2082.44\,{{ q_{2}}}^{3} \alpha _{{3}}\alpha _{{1}}{ q_{1}} - 1022.84\,{{ q_{2}}}^{4}\alpha _{{3}} \alpha _{{1}}{ q_{1}}\\ n_{9o}= & {} 2.65\,\beta _{{3\,g}}{{ q_{2}}}^{4}- 7.41\,\beta _{{3\,g}}{{ q_{2}}}^{3 }\\&+\, 7.98\,\beta _{{3\,g}}{{ q_{2}}}^{2}- 4.0\,\beta _{{3\,g}}{ q_{2}} + 0.83\,\beta _{{3\,g}}\\&-\, 0.83\,\beta _{{g}}+ 1.0\,\beta _{{g}}{ q_{2}}\\ n_{10o}= & {} - 1.85\,\alpha _{{g}}{{ q_{2}}}^{3}+ 1.0\,\alpha _{{2\,g}}{ q_{2}}- 2.55 \,\alpha _{{2\,g}}{{ q_{2}}}^{4}\\&-\, 3.0\,\alpha _{{g}}{ q_{2}}+ 0.83\, \alpha _{{g}}\\&+\, 5.56 \alpha _{{2\,g}}{{ q_{2}}}^{3}-3.98\,\alpha _{{2\,g }}{{ q_{2}}}^{2}\\&+\, 3.98\,\alpha _{{g}}{{ q_{2}}}^{2}\\ n_{11o}= & {} - 3.98\,{ q2}\,c_{{3}}- 2.65\,{{ q2}}^{3}c_{{3}}+ 1.0\,c_{{2}}\\&+\, 5.56\,{{ q2}}^{2}c_{{3}} \end{aligned}$$

$$\begin{aligned} \lambda _{2}= & {} \sqrt{\frac{k_{o}}{m_{o}}},\quad s_{1}=\frac{n_{1o}}{m_{o}},\quad s_{2}=\frac{n_{2o}}{m_{o}}, \\ s_{3}= & {} \frac{n_{3o}}{m_{o}},\quad s_{4}=\frac{n_{4o}}{m_{o}},\quad s_{5}=\frac{n_{5o}}{m_{o}}, \\ s_{6}= & {} \frac{n_{6o}}{m_{o}},\quad s_{7}=\frac{n_{7o}}{m_{o}},\quad s_{8}=\frac{n_{8o}}{m_{o}}, \\ s_{9}= & {} \frac{n_{9o}}{m_{o}},\quad s_{10}=\frac{n_{10o}}{m_{o}},\quad s_{11}=\frac{n_{11o}}{m_{o}} \end{aligned}$$

Appendix 2

$$\begin{aligned} B_{1}= & {} \left( 2\,k_{{2}}t_{{11}}-2\,k_{{1}}s_{{11}} \right) \omega _{{1}},\\ B_{2}= & {} \left( 2\,k_{{2}}t_{{11}}-2\,k_{{2}}s_{{11}} \right) \omega _{{2}}, \\ B_{3}= & {} s_{{9}}\eta _{{21}}+s_{{10}} ( \eta _{{11}} +\eta _{{12}} ) -k_ {{2}} \left( t_{{9}}\eta _{{11}}+t_{{10}}\eta _{{12}} \right) \\ B_{4}= & {} 4\,\omega _{{1}} \left( k_{{1}}-k_{{2}} \right) \quad B_{5}=\left( 2\,k_{{2}}s_{{11}}-2\,k_{{1}}t_{{11}} \right) \omega _{{2}} , \\ B_{6}= & {} \left( 2\,k_{{1}}s_{{11}}-2\,k_{{1}}t_{{11}} \right) \omega _{{1}}, \\ B_{7}= & {} k_{{1}} \left( t_{{9}}\eta _{{11}}+t_{{10}}\eta _{{12}} \right) -s_{{9}} \eta _{{21}}-s_{{10}} \left( \eta _{{11}}+\eta _{{12}} \right) ,\\ B_{8}= & {} 4\,\omega _{{2}} \left( k_{{1}}-k_{{2}} \right) . \end{aligned}$$

$$\begin{aligned} c_{11}= & {} -{\frac{-{\lambda _{{2}}}^{2}t_{{2}}+t_{{8}}s_{{2}}{k_{{1}}}^{2}+t_{{8 }}s_{{4}}k_{{1}}+t_{{8}}s_{{7}}+4\,{\omega _{{1}}}^{2}t_{{2}}+4\,{ \omega _{{1}}}^{2}t_{{4}}k_{{1}}+4\,{\omega _{{1}}}^{2}t_{{7}}{k_{{1}}}^ {2}-{\lambda _{{2}}}^{2}t_{{7}}{k_{{1}}}^{2}-{\lambda _{{2}}}^{2}t_{{4}} k_{{1}}}{t_{{8}}s_{{8}}-16\,{\omega _{{1}}}^{4}+4\,{\omega _{{1}}}^{2}{ \lambda _{{1}}}^{2}+4\,{\lambda _{{2}}}^{2}{\omega _{{1}}}^{2}-{\lambda _{ {2}}}^{2}{\lambda _{{1}}}^{2}}} \\ c_{12}= & {} -2\,{\frac{-{\lambda _{{2}}}^{2}t_{{2}}+t_{{8}}s_{{2}}{k_{{1}}}^{2}+t_ {{8}}s_{{4}}k_{{1}}+t_{{8}}s_{{7}}-{\lambda _{{2}}}^{2}t_{{4}}k_{{1}}-{ \lambda _{{2}}}^{2}t_{{7}}{k_{{1}}}^{2}}{t_{{8}}s_{{8}}-{\lambda _{{2}}} ^{2}{\lambda _{{1}}}^{2}}} \\ c_{13}= & {} -{\frac{-{\lambda _{{2}}}^{2}t_{{2}}+t_{{8}}s_{{2}}{k_{{2}}}^{2}+t_{{8 }}s_{{4}}k_{{2}}+t_{{8}}s_{{7}}+4\,{\omega _{{2}}}^{2}t_{{2}}+4\,{ \omega _{{2}}}^{2}t_{{4}}k_{{2}}+4\,{\omega _{{2}}}^{2}t_{{7}}{k_{{2}}}^ {2}-{\lambda _{{2}}}^{2}t_{{7}}{k_{{2}}}^{2}-{\lambda _{{2}}}^{2}t_{{4}} k_{{2}}}{t_{{8}}s_{{8}}-16\,{\omega _{{2}}}^{4}+4\,{\omega _{{2}}}^{2}{ \lambda _{{1}}}^{2}+4\,{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-{\lambda _{ {2}}}^{2}{\lambda _{{1}}}^{2}}} \\ c_{14}= & {} -2\,{\frac{-{\lambda _{{2}}}^{2}t_{{2}}+t_{{8}}s_{{2}}{k_{{2}}}^{2}+t_ {{8}}s_{{4}}k_{{2}}+t_{{8}}s_{{7}}-{\lambda _{{2}}}^{2}t_{{4}}k_{{2}}-{ \lambda _{{2}}}^{2}t_{{7}}{k_{{2}}}^{2}}{t_{{8}}s_{{8}}-{\lambda _{{2}}} ^{2}{\lambda _{{1}}}^{2}}}, \\ c_{15}= & {} \frac{c_{15N}}{c_{15D}}, \\ c_{16}= & {} \frac{c_{16N}}{c_{16D}} \end{aligned}$$

$$\begin{aligned} c_{15N}= & {} - 2\,t_{{8}}s_{{2}}k_{{1}}k_{{2}}+2\,{\omega _{{1}}}^{2}t_{{7}}k _{{1}}k_{{2}}\\&+\,{\omega _{{2}}}^{2}t_{{4}}k_{{2}}-2\,\omega _{{1}}\omega _{ {2}}t_{{4}}k_{{1}}-2\,\omega _{{1}}\omega _{{2}}t_{{4}}k_{{2}}\\&+\,2\,{ \omega _{{2}}}^{2} t_{{7}}k_{{1}}k_{{2}}+\,2\,{\omega _{{1}}}^{2}t_{{2}}\\&-\,2 \,{\lambda _{{2}}}^{2}t_{{7}}k_{{1}}k_{{2}}\\&-\,4\,\omega _{{1}}\omega _{{2}} t_{{7}}k_{{1}}k_{{2}}-{\lambda _{{2}}}^{2}t_{{4}}k_{{2}}\\&+\,t_{{8}}s_{{4}} k_{{1}} - {\lambda _{{2}}}^{2}t_{{4}}k_{{1}}\\&+\,t_{{8}}s_{{4}}k_{{2}}+2\,t_{ {8}}s_{{7}}+{\omega _{{1}}}^{2}t_{{4}}k_{{1}}\\&-\,4\,\omega _{{1}}\omega _{{2 }}t_{{2}}+{\omega _{{2}}}^{2}t_{{4}}k_{{1}} \\&+\, {\omega _{{1}}}^{2}t_{{4}}k_ {{2}}-2\,{\lambda _{{2}}}^{2}t_{{2}}\\&+\,2\,{\omega _{{2}}}^{2}t_{{2}}\\ c_{15D}= & {} - 2\,\omega _{{1}}\omega _{{2}}{\lambda _{{1}}}^{2}-2\,{\lambda _{{2}}}^{2} \omega _{{1}}\omega _{{2}}\\&+\,4\,{\omega _{{1}}}^{3}\omega _{{2}}+t_{{8}}s_{{ 8}}-{\omega _{{2}}}^{4}\\&-\,{\omega _{{1}}}^{4}-6\,{\omega _{{1}}}^{2}{\omega _{{2}}}^{2} \\&+\, {\omega _{{1}}}^{2}{\lambda _{{1}}}^{2}+4\,\omega _{{1}}{ \omega _{{2}}}^{3}\\&+\,{\omega _{{2}}}^{2}{\lambda _{{1}}}^{2}+{\lambda _{{2}} }^{2}{\omega _{{1}}}^{2}\\&+\,{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-{\lambda _{{2}}}^{2}{\lambda _{{1}}}^{2}\\ c_{16N}= & {} -2\,t_{{8}}s_{{2}}k_{{1}}k_{{2}}+2\,{\omega _{{1}}}^{2}t_{{7}}k _{{1}}k_{{2}}\\&+\,2\,\omega _{{1}}\omega _{{2}}t_{{4}}k_{{1}}+2\,\omega _{{1} }\omega _{{2}}t_{{4}}k_{{2}}\\&+\,2\,{\omega _{{2}}}^{2}t_{{7}}k_{{1}}k_{{2}}-\, 2\,{\lambda _{{2}}}^{2}t_{{7}}k_{{1}}k_{{2}} \\&-\,{\lambda _{{2}}}^{2}t_{{4} }k_{{1}}+t_{{8}}s_{{4}}k_{{2}} +\,4\,\omega _{{1}}\omega _{{2}}t_{{7}}k_{{1 }}k_{{2}}\\&+\,t_{{8}}s_{{4}}k_{{1}}+{\omega _{{2}}}^{2}t_{{4}}k_{{2}}+2\,t_ {{8}}s_{{7}}\\&-\,{\lambda _{{2}}}^{2}t_{{4}}k_{{2}}+{\omega _{{1}}}^{2}t_{{4 }}k_{{1}}+2\,{\omega _{{2}}}^{2}t_{{2}} \\&+\,{\omega _{{2}}}^{2}t_{{4}}k_{{1} } - 2\,{\lambda _{{2}}}^{2}t_{{2}}\\&+\,2\,{\omega _{{1}}}^{2}t_{{2}}+4\,\omega _{{1}}\omega _{{2}}t_{{2}}+{\omega _{{1}}}^{2}t_{{4}}k_{{2}} \end{aligned}$$

$$\begin{aligned} c_{16D}= & {} 2\,\omega _ {{1}}\omega _{{2}}{\lambda _{{1}}}^{2}+2\,{\lambda _{{2}}}^{2}\omega _{{1} }\omega _{{2}}\\&-\,{\omega _{{1}}}^{4}-{\omega _{{2}}}^{4}+t_{{8}}s_{{8}}-4\, {\omega _{{1}}}^{3}\omega _{{2}}\\&-\,6\,{\omega _{{1}}}^{2}{\omega _{{2}}}^{2} +{\omega _{{1}}}^{2}{\lambda _{{1}}}^{2}-4\,\omega _{{1}}{\omega _{{2}}}^{ 3}\\&+\,{\omega _{{2}}}^{2}{\lambda _{{1}}}^{2}+{\lambda _{{2}}}^{2}{\omega _{{ 1}}}^{2}\\&+\,{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-{\lambda _{{2}}}^{2}{ \lambda _{{1}}}^{2} \end{aligned}$$

$$\begin{aligned} c_{21}= & {} {\frac{{\lambda _{{1}}}^{2}s_{{7}}-t_{{2}}s_{{8}}-4\,{\omega _{{1}}}^{2 }s_{{7}}-4\,{\omega _{{1}}}^{2}s_{{4}}k_{{1}}+{\lambda _{{1}}}^{2}s_{{4} }k_{{1}}-t_{{4}}k_{{1}}s_{{8}}-4\,{\omega _{{1}}}^{2}s_{{2}}{k_{{1}}}^{ 2}+{\lambda _{{1}}}^{2}s_{{2}}{k_{{1}}}^{2}-t_{{7}}{k_{{1}}}^{2}s_{{8}} }{t_{{8}}s_{{8}}-16\,{\omega _{{1}}}^{4}+4\,{\omega _{{1}}}^{2}{\lambda _ {{1}}}^{2}+4\,{\lambda _{{2}}}^{2}{\omega _{{1}}}^{2}-{\lambda _{{2}}}^{2 }{\lambda _{{1}}}^{2}}} \\ c_{22}= & {} 2\,{\frac{{\lambda _{{1}}}^{2}s_{{2}}{k_{{1}}}^{2}+{\lambda _{{1}}}^{2} s_{{4}}k_{{1}}+{\lambda _{{1}}}^{2}s_{{7}}-t_{{2}}s_{{8}}-t_{{4}}k_{{1} }s_{{8}}-t_{{7}}{k_{{1}}}^{2}s_{{8}}}{t_{{8}}s_{{8}}-{\lambda _{{2}}}^{ 2}{\lambda _{{1}}}^{2}}} \\ c_{23}= & {} {\frac{-4\,{\omega _{{2}}}^{2}s_{{7}}+{\lambda _{{1}}}^{2}s_{{7}}-t_{{2 }}s_{{8}}-4\,{\omega _{{2}}}^{2}s_{{4}}k_{{2}}+{\lambda _{{1}}}^{2}s_{{4 }}k_{{2}}-t_{{4}}k_{{2}}s_{{8}}-4\,{\omega _{{2}}}^{2}s_{{2}}{k_{{2}}}^ {2}+{\lambda _{{1}}}^{2}s_{{2}}{k_{{2}}}^{2}-t_{{7}}{k_{{2}}}^{2}s_{{8} }}{t_{{8}}s_{{8}}-16\,{\omega _{{2}}}^{4}+4\,{\omega _{{2}}}^{2}{\lambda _{{1}}}^{2}+4\,{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-{\lambda _{{2}}}^{ 2}{\lambda _{{1}}}^{2}}} \\ c_{24}= & {} 2\,{\frac{{\lambda _{{1}}}^{2}s_{{2}}{k_{{2}}}^{2}+{\lambda _{{1}}}^{2} s_{{4}}k_{{2}}+{\lambda _{{1}}}^{2}s_{{7}}-t_{{2}}s_{{8}}-t_{{4}}k_{{2} }s_{{8}}-t_{{7}}{k_{{2}}}^{2}s_{{8}}}{t_{{8}}s_{{8}}-{\lambda _{{2}}}^{ 2}{\lambda _{{1}}}^{2}}},\\ c_{25}= & {} \frac{c_{25N}}{c_{25D}}, \\ c_{26}= & {} \frac{c_{26N}}{c_{26D}} \end{aligned}$$

$$\begin{aligned} c_{25N}= & {} - 2\,{\omega _{{1}}}^{2}s_{{2}}k_{{1}}k_{{2}}+2\,\omega _{{1}} \omega _{{2}}s_{{4}}k_{{2}}+2\,\omega _{{1}}\omega _{{2}}s_{{4}}k_{{1}}\\&-\,2 \,{\omega _{{2}}}^{2}s_{{2}}k_{{1}}k_{{2}}+2\,{\lambda _{{1}}}^{2}s_{{2} }k_{{1}}k_{{2}}- 2\,t_{{7}}k_{{1}}k_{{2}}s_{{8}}\\&-\,2\,{\omega _{{2}}}^{2}s _{{7}}+2\,{\lambda _{{1}}}^{2}s_{{7}}-2\,t_{{2}}s_{{8}}-2\,{\omega _{{1} }}^{2}s_{{7}}\\&-{\omega _{{1}}}^{2}s_{{4}}k_{{2}}-\,{\omega _{{1}}}^{2}s_{{4 }}k_{{1}}+4\,\omega _{{1}}\omega _{{2}}s_{{7}}\\&-\,{\omega _{{2}}}^{2}s_{{4}} k_{{2}}-{\omega _{{2}}}^{2}s_{{4}}k_{{1}}+{\lambda _{{1}}}^{2}s_{{4}}k_{ {2}}+{\lambda _{{1}}}^{2}s_{{4}}k_{{1}}\\&-\,t_{{4}}k_{{1}}s_{{8}}-t_{{4}}k_ {{2}}s_{{8}}+4\,\omega _{{1}}\omega _{{2}}s_{{2}}k_{{1}}k_{{2}}\\ c_{25D}= & {} - 2\,\omega _{{1}}\omega _{{2}}{\lambda _{{1}}}^{2}-2\,{\lambda _{{2}}}^{2} \omega _{{1}}\omega _{{2}}\\&+\,4\,{\omega _{{1}}}^{3}\omega _{{2}}+t_{{8}}s_{{ 8}}-{\omega _{{2}}}^{4}-{\omega _{{1}}}^{4}\\&-\,6\,{\omega _{{1}}}^{2}{\omega _{{2}}}^{2}+{\omega _{{1}}}^{2}{\lambda _{{1}}}^{2}+4\,\omega _{{1}}{ \omega _{{2}}}^{3}\\&+\,{\omega _{{2}}}^{2}{\lambda _{{1}}}^{2}+{\lambda _{{2}} }^{2}{\omega _{{1}}}^{2}\\&+\,{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-{\lambda _{{2}}}^{2}{\lambda _{{1}}}^{2} \\ c_{26N}= & {} - 2\,{\omega _{{1}}}^{2}s_{{2}}k_{{1}}k_{{2}}-2\,\omega _{{1}} \omega _{{2}}s_{{4}}k_{{2}}-2\,\omega _{{1}}\omega _{{2}}s_{{4}}k_{{1}}\\&-\,2 \,{\omega _{{2}}}^{2}s_{{2}}k_{{1}}k_{{2}}+2\,{\lambda _{{1}}}^{2}s_{{2} }k_{{1}}k_{{2}} \\&-\, 2\,t_{{7}}k_{{1}}k_{{2}}s_{{8}}-2\,{\omega _{{2}}}^{2}s _{{7}}+2\,{\lambda _{{1}}}^{2}s_{{7}}\\&-\,2\,t_{{2}}s_{{8}}-2\,{\omega _{{1} }}^{2}s_{{7}}-{\omega _{{1}}}^{2}s_{{4}}k_{{2}}-{\omega _{{1}}}^{2}s_{{4 }}k_{{1}} \\&-\, 4\,\omega _{{1}}\omega _{{2}}s_{{7}}-{\omega _{{2}}}^{2}s_{{4}} k_{{2}}\\&-\,{\omega _{{2}}}^{2}s_{{4}}k_{{1}}+{\lambda _{{1}}}^{2}s_{{4}}k_{ {2}}\\&+\,{\lambda _{{1}}}^{2}s_{{4}}k_{{1}}-t_{{4}}k_{{1}}s_{{8}}-t_{{4}}k_ {{2}}s_{{8}} \\&-\, 4\,\omega _{{1}}\omega _{{2}}s_{{2}}k_{{1}}k_{{2}}\\ c_{26D}= & {} 2\,\omega _{{1}}\omega _{{2}}{\lambda _{{1}}}^{2}+2\,{\lambda _{{2}}}^{2} \omega _{{1}}\omega _{{2}}-\,{\omega _{{1}}}^{4}\\&-\,{\omega _{{2}}}^{4}+t_{{8}} s_{{8}}-4\,{\omega _{{1}}}^{3}\omega _{{2}}-\,6\,{\omega _{{1}}}^{2}{\omega _{{2}}}^{2}\\&+\, {\omega _{{1}}}^{2}{\lambda _{{1}}}^{2} -4\,\omega _{{1}}{ \omega _{{2}}}^{3}+\,{\omega _{{2}}}^{2}{\lambda _{{1}}}^{2}\\&+\,{\lambda _{{2}} }^{2}{\omega _{{1}}}^{2}+{\lambda _{{2}}}^{2}{\omega _{{2}}}^{2}-\,{\lambda _{{2}}}^{2}{\lambda _{{1}}}^{2} \end{aligned}$$

$$\begin{aligned} g_{11}= & {} 2\,t_{{2}}c_{{15}}+2\,t_{{2}}c_{{11}}+t_{{3}}{k_{{1}}}^{2}+t_{{4}}c_{{ 21}}\\&+\,t_{{4}}c_{{25}}+t_{{4}}k_{{1}}c_{{15}}+2\,t_{{3}}k_{{1}}k_{{2}}+t _{{4}}k_{{2}}c_{{11}}\\&+\,2\,t_{{7}}k_{{1}}c_{{25}}+2\,t_{{7}}k_{{2}}c_{{ 21}}+3\,t_{{1}}\\ g_{12}= & {} t_{{4}}k_{{2}}c_{{13}}+3\,t_{{1}}+2\,t_{{2}}c_{{13}}+3\,t_{{3}}{k_{{2} }}^{2}\\&+\,2\,t_{{2}}c_{{14}}+t_{{4}}k_{{2}}c_{{14}}+2\,t_{{7}}k_{{2}}c_{{ 24}}\\&+\,2\,t_{{7}}k_{{2}}c_{{23}}+t_{{4}}c_{{24}}+t_{{4}}c_{{23}}\\ g_{13}= & {} 2\,t_{{7}}k_{{2}}c_{{22}}+t_{{4}}c_{{22}}+t_{{4}}c_{{26}}+t_{{4}}k_{{2 }}c_{{12}}\\&+\,2\,t_{{2}}c_{{16}}+t_{{4}}k_{{1}}c_{{16}}+2\,t_{{2}}c_{{12} }\\&+\,4\,t_{{3}}k_{{1}}k_{{2}}+2\,t_{{3}}{k_{{1}}}^{2}+6\,t_{{1}}+2\,t_{{7 }}k_{{1}}c_{{26}}\\ g_{14}= & {} 2\,t_{{2}}c_{{13}}+t_{{3}}{k_{{2}}}^{2}+t_{{4}}c_{{23}}+2\,t_{{3}}k_{{ 1}}k_{{2}}\\&+\,t_{{4}}k_{{1}}c_{{13}}+2\,t_{{7}}k_{{1}}c_{{23}}+3\,t_{{1}}\\ g_{15}= & {} 2\,t_{{7}}k_{{1}}c_{{22}}+2\,t_{{7}}k_{{1}}c_{{21}}+t_{{4}}k_{{1}}c_{{ 12}}\\&+\,2\,t_{{2}}c_{{12}}+t_{{4}}c_{{22}}+t_{{4}}k_{{1}}c_{{11}}+3\,t_{{ 3}}{k_{{1}}}^{2}\\&+\,t_{{4}}c_{{21}}+3\,t_{{1}}+2\,t_{{2}}c_{{11}}\\ g_{16}= & {} t_{{4}}k_{{1}}c_{{14}}+2\,t_{{7}}k_{{1}}c_{{24}}+t_{{4}}k_{{2}}c_{{15} }\\&+\,t_{{4}}k_{{2}}c_{{16}}+2\,t_{{7}}k_{{2}}c_{{25}}\\&+\,4\,t_{{3}}k_{{1}}k_ {{2}}+2\,t_{{7}}k_{{2}}c_{{26}}+2\,t_{{3}}{k_{{2}}}^{2}\\&+\,t_{{4}}c_{{25} }+t_{{4}}c_{{26}}+2\,t_{{2}}c_{{14}}+2\,t_{{2}}c_{{16}}\\&+\,t_{{4}}c_{{24} }+2\,t_{{2}}c_{{15}}+6\,t_{{1}}\\ f_{11}= & {} 2\,t_{{2}}c_{{13}}+t_{{3}}{k_{{2}}}^{2}+t_{{4}}c_{{23}}+2\,t_{{3}}k_{{ 1}}k_{{2}}\\&+\,t_{{4}}k_{{1}}c_{{13}}+2\,t_{{7}}k_{{1}}c_{{23}}+3\,t_{{1}}\\ f_{12}= & {} 2\,t_{{7}}k_{{1}}c_{{22}}+2\,t_{{7}}k_{{1}}c_{{21}}+t_{{4}}k_{{1}}c_{{ 12}}\\&+\,2\,t_{{2}}c_{{12}}+t_{{4}}c_{{22}}+t_{{4}}k_{{1}}c_{{11}}+3\,t_{{ 3}}{k_{{1}}}^{2}\\&+\,t_{{4}}c_{{21}}+3\,t_{{1}}+2\,t_{{2}}c_{{11}}\\ f_{13}= & {} t_{{4}}k_{{1}}c_{{14}}+2\,t_{{7}}k_{{1}}c_{{24}}+t_{{4}}k_{{2}}c_{{15} }\\&+\,t_{{4}}k_{{2}}c_{{16}}+2\,t_{{7}}k_{{2}}c_{{25}}+4\,t_{{3}}k_{{1}}k_ {{2}}\\&+\,2\,t_{{7}}k_{{2}}c_{{26}}+2\,t_{{3}}{k_{{2}}}^{2}+t_{{4}}c_{{25} }\\&+\,t_{{4}}c_{{26}}+2\,t_{{2}}c_{{14}}+2\,t_{{2}}c_{{16}}+t_{{4}}c_{{24} }\\&+\,2\,t_{{2}}c_{{15}}+6\,t_{{1}} \end{aligned}$$

$$\begin{aligned} f_{14}= & {} 2\,t_{{2}}c_{{15}}+2\,t_{{2}}c_{{11}}+t_{{3}}{k_{{1}}}^{2}+t_{{4}}c_{{ 21}}\\&+\,t_{{4}}c_{{25}}+t_{{4}}k_{{1}}c_{{15}}+2\,t_{{3}}k_{{1}}k_{{2}}+t _{{4}}k_{{2}}c_{{11}}\\&+\, 2\,t_{{7}}k_{{1}}c_{{25}}+2\,t_{{7}}k_{{2}}c_{{ 21}}+3\,t_{{1}} \\ f_{15}= & {} t_{{4}}k_{{2}}c_{{13}}+3\,t_{{1}}+2\,t_{{2}}c_{{13}}+3\,t_{{3}}{k_{{2} }}^{2}\\&+\,2\,t_{{2}}c_{{14}}+t_{{4}}k_{{2}}c_{{14}}+2\,t_{{7}}k_{{2}}c_{{ 24}}+2\,t_{{7}}k_{{2}}c_{{23}} \\&+\, t_{{4}}c_{{24}}+t_{{4}}c_{{23}} \\ f_{16}= & {} 2\,t_{{7}}k_{{2}}c_{{22}}+t_{{4}}c_{{22}}+t_{{4}}c_{{26}}+t_{{4}}k_{{2 }}c_{{12}}\\&+\,2\,t_{{2}}c_{{16}}+t_{{4}}k_{{1}}c_{{16}}+2\,t_{{2}}c_{{12} }+4\,t_{{3}}k_{{1}}k_{{2}}\\&+\, 2\,t_{{3}}{k_{{1}}}^{2}+6\,t_{{1}}+2\,t_{{7 }}k_{{1}}c_{{26}} \end{aligned}$$

$$\begin{aligned} g_{21}= & {} s_{{4}}c_{{21}}+s_{{3}}k_{{2}}+2\,s_{{3}}k_{{1}}+s_{{4}}c_{{25}}+2\,s_ {{7}}c_{{15}}\\&+\,3\,s_{{1}}{k_{{1}}}^{2}k_{{2}}+s_{{4}}c_{{15}}k_{{1}}+2 \,s_{{2}}k_{{1}}c_{{25}}2\,s_{{2}}k_{{2}}c_{{21}}\\&+\,2\,s_{{7}}c_{{11}}+ s_{{4}}c_{{11}}k_{{2}}\\ g_{22}= & {} 2\,s_{{2}}k_{{2}}c_{{24}}+3\,s_{{3}}k_{{2}}+2\,s_{{7}}c_{{14}}+s_{{4}} c_{{24}}\\&+\,s_{{4}}c_{{23}}+s_{{4}}c_{{13}}k_{{2}}+2\,s_{{2}}k_{{2}}c_{{ 23}}\\&+\,s_{{4}} c_{{14}}k_{{2}}+2\,s_{{7}}c_{{13}}+3\,s_{{1}}{k_{{2}}}^{3}\\ g_{23}= & {} s_{{4}}c_{{12}}k_{{2}}+2\,s_{{3}}k_{{2}}+2\,s_{{7}}c_{{12}}+6\,s_{{1}} {k_{{1}}}^{2}k_{{2}}\\&+\,2\,s_{{7}}c_{{16}}+s_{{4}}c_{{22}}+4\,s_{{3}}k_{{ 1}}+2\,s_{{2}} k_{{1}}c_{{26}}\\&+\,s_{{4}}c_{{26}}+2\,s_{{2}}k_{{2}}c_{{22} }+s_{{4}}c_{{16}}k_{{1}}\\ g_{24}= & {} 2\,s_{{2}}k_{{1}}c_{{23}}+3\,s_{{1}}{k_{{2}}}^{2}k_{{1}}+s_{{4}}c_{{13 }}k_{{1}}\\&+\,s_{{4}}c_{{23}}+2\,s_{{7}}c_{{13}}+s_{{3}}k_{{1}}+2\,s_{{3}} k_{{2}}\\ g_{25}= & {} 2\,s_{{7}}c_{{12}}+2\,s_{{2}}k_{{1}}c_{{22}}+3\,s_{{3}}k_{{1}}+s_{{4}} c_{{12}}k_{{1}}\\&+\,s_{{4}}c_{{11}}k_{{1}}+2\,s_{{2}}k_{{1}}c_{{21}}+s_{{4 }}c_{{22}}s_{{4}}c_{{21}}\\&+\,2\,s_{{7}}c_{{11}}+3\,s_{{1}}{k_{{1}}}^{3}\\ g_{26}= & {} s_{{4}}c_{{14}}k_{{1}}+6\,s_{{1}}{k_{{2}}}^{2}k_{{1}}+2\,s_{{2}}k_{{1} }c_{{24}}\\&+\,s_{{4}}c_{{15}}k_{{2}}+s_{{4}}c_{{16}}k_{{2}}+2\,s_{{2}}k_{{ 2}}c_{{25}}2\,s_{{2}}k_{{2}}c_{{26}}\\&+\,2\,s_{{7}}c_{{14}}+2\,s_{{7}}c_{ {15}}+4\,s_{{3}}k_{{2}}+2\,s_{{3}}k_{{1}}\\&+\,2\,s_{{7}}c_{{16}}+s_{{4}}c_ {{24}}s_{{4}}c_{{26}}+s_{{4}}c_{{25}}\\ f_{21}= & {} 2\,s_{{2}}k_{{1}}c_{{23}}+3\,s_{{1}}{k_{{2}}}^{2}k_{{1}}+s_{{4}}c_{{13 }}k_{{1}}\\&+\,s_{{4}}c_{{23}}+2\,s_{{7}}c_{{13}}+s_{{3}}k_{{1}}+2\,s_{{3}} k_{{2}}\\ f_{22}= & {} 2\,s_{{7}}c_{{12}}+2\,s_{{2}}k_{{1}}c_{{22}}+3\,s_{{3}}k_{{1}}+s_{{4}} c_{{12}}k_{{1}}\\&+\,s_{{4}}c_{{11}}k_{{1}}+2\,s_{{2}}k_{{1}}c_{{21}}+s_{{4 }}c_{{22}}s_{{4}}c_{{21}}\\&+\,2\,s_{{7}}c_{{11}}+3\,s_{{1}}{k_{{1}}}^{3}\\ f_{23}= & {} s_{{4}}c_{{14}}k_{{1}}+6\,s_{{1}}{k_{{2}}}^{2}k_{{1}}+2\,s_{{2}}k_{{1} }c_{{24}}\\&+\,s_{{4}}c_{{15}}k_{{2}}+s_{{4}}c_{{16}}k_{{2}}+2\,s_{{2}}k_{{ 2}}c_{{25}}2\,s_{{2}}k_{{2}}c_{{26}}\\&+\,2\,s_{{7}}c_{{14}}+2\,s_{{7}}c_{ {15}}\\&+\,4\,s_{{3}}k_{{2}}+2\,s_{{3}}k_{{1}}+2\,s_{{7}}c_{{16}}+s_{{4}}c_ {{24}}s_{{4}}c_{{26}}+s_{{4}}c_{{25}}\\ f_{24}= & {} s_{{4}}c_{{21}}+s_{{3}}k_{{2}}+2\,s_{{3}}k_{{1}}+s_{{4}}c_{{25}}+2\,s_ {{7}}c_{{15}}\\&+\,3\,s_{{1}}{k_{{1}}}^{2}k_{{2}}+s_{{4}}c_{{15}}k_{{1}}2 \,s_{{2}}k_{{1}}c_{{25}}\\&+\,2\,s_{{2}}k_{{2}}c_{{21}}+2\,s_{{7}}c_{{11}}\\&+\, s_{{4}}c_{{11}}k_{{2}}\\ f_{25}= & {} 2\,s_{{2}}k_{{2}}c_{{24}}+3\,s_{{3}}k_{{2}}+2\,s_{{7}}c_{{14}}+s_{{4}} c_{{24}}\\&+\,s_{{4}}c_{{23}}+s_{{4}}c_{{13}}k_{{2}}+2\,s_{{2}}k_{{2}}c_{{ 23}}s_{{4}}c_{{14}}k_{{2}}+2\,s_{{7}}c_{{13}}\\&+\,3\,s_{{1}}{k_{{2}}}^{3}\\ f_{26}= & {} s_{{4}}c_{{12}}k_{{2}}+2\,s_{{3}}k_{{2}}+2\,s_{{7}}c_{{12}}+6\,s_{{1}} {k_{{1}}}^{2}k_{{2}}\\&+\,2\,s_{{7}}c_{{16}}+s_{{4}}c_{{22}}+4\,s_{{3}}k_{{ 1}}2\,s_{{2}}k_{{1}}c_{{26}}\\&+\,s_{{4}}c_{{26}}2\,s_{{2}}k_{{2}}c_{{22} }+s_{{4}}c_{{16}}k_{{1}} \end{aligned}$$

$$\begin{aligned} B_{11}= & {} 1+k_{{1}}\bar{k_{{1}}}, \quad B_{12}=1+k_{{2}}\bar{k_{{1}}},\\ B_{13}= & {} t_{{11}}+s_{{11}}k_{{1}}\bar{k_{{1}}},\\ B_{14}= & {} t_{{11}}+s_{{11}}k_{{2}}\bar{k_{{1}}}, \quad G_{11} = 1+k_{{1}}\bar{k_{{2}}}, \\ G_{12}= & {} 1+k_{{2}}\bar{k_{{2}}}, \\ G_{13}= & {} t_{{11}}+s_{{11}}k_{{2}}\bar{k_{{2}}}, G_{14}=t_{{11}}+s_{{11}}k_{{1}}\bar{k_{{2}}}\\ g_{1}= & {} g_{{11}}+\bar{k_{{1}}}g_{{21}}, \quad g_{2}=g_{{12}}+\bar{k_{{1}}}g_{{22}}, \\ g_{3}= & {} g_{{13}}+\bar{k_{{1}}}g_{{23}}, \\ g_{4}= & {} g_{{14}}+\bar{k_{{1}}}g_{{24}}, \quad g_{5}=g_{{15}}+\bar{k_{{1}}}g_{{25}}, \\ g_{6}= & {} g_{{16}}+\bar{k_{{1}}}g_{{26}}, \\ f_{1}= & {} f_{{11}}+\bar{k_{{2}}}f_{{21}}, \quad f_{2}=f_{{12}}+\bar{k_{{2}}}f_{{22}}, \\ f_{3}= & {} f_{{13}}+\bar{k_{{2}}}f_{{23}},\\ f_{4}= & {} f_{{14}}+\bar{k_{{2}}}f_{{24}}, \quad f_{5}=f_{{15}}+\bar{k_{{2}}}f_{{25}}, \\ f_{6}= & {} f_{{16}}+\bar{k_{{2}}}f_{{26}} \end{aligned}$$

$$\begin{aligned} \bar{G}= & {} \bar{g_{1}}\bar{A_{2}}A_{1}^2e^{-i\sigma _{2}T_{1}}+\bar{g_{2}}\bar{A_{2}}A_{2}^2e^{i\sigma _{2}T_{1}}\nonumber \\&+\bar{g_{3}}A_{1}A_{2}\bar{A_{1}}e^{i\sigma _{2}T_{1}}\\&+\,\bar{g_{4}}A_{2}^2\bar{A_{1}}e^{2i\sigma _{2}T_{1}} + \bar{g_{5}}A_{1}^2\bar{A_{1}}+\bar{g_{6}}A_{1}\bar{A_{2}}A_{2}; \\ \bar{F}= & {} \bar{f_{1}}\bar{A_{1}}A_{2}^2e^{i\sigma _{2}T_{1}}+\bar{f_{2}}\bar{A_{1}}A_{1}^2e^{-i\sigma _{2}T_{1}}\\&+\,\bar{f_{3}}A_{1}A_{2}\bar{A_{2}}e^{-i\sigma _{2}T_{1}}+\bar{f_{4}}A_{1}^2\bar{A_{2}}e^{-2i\sigma _{2}T_{1}} \\&+\, \bar{f_{5}}A_{2}^2\bar{A_{2}}+\bar{f_{6}}A_{1}\bar{A_{1}}A_{2}; \end{aligned}$$

$$\begin{aligned}&h_{11}=1/4\,{\frac{B_{{13}}G_{{12}}-B_{{12}}G_{{14}}}{\omega _{{1}} \left( B_ {{11}}G_{{12}}-B_{{12}}G_{{11}} \right) }}, \\&h_{22}=1/2\,{\frac{B_{{1}}}{B_{{4}}}}, \quad h_{33}=1/2\,{\frac{B_{{2}}}{B_{{4}}}},\\&h_{44}={\frac{B_{{3}}}{B_{{4}}}}, \\&h_{55}=1/4\,{\frac{B_{{14}}G_{{12}}-B_{{12}}G_{{13}}}{\omega _{{1}} \left( B_{{11}}G_{{12}}-B_{{12}}G_{{11}} \right) }}, \\&h_{66}=1/2\,{\frac{B_{{5}}}{B_{{8}}}}, \quad h_{77}=1/2\,{\frac{B_{{6}}}{B_{{8}}}}, \\&h_{88}={\frac{B_{{7}}}{B_{{8}}}}, \\&h_{99}=1/16\,{\frac{G_{{12}}}{\omega _{{1}} \left( B_{{11}}G_{{12}}-B_{{12}}G_{{11}} \right) }}, \\&h_{1010}=1/16\,{\frac{B_{{12}}}{\omega _{{1}} \left( B_{{11}}G_{{12}}-B_{{12}}G _{{11}} \right) }} \\&h_{1}=h_{11},\quad h_{2}=2h_{22},\quad h_{3}=2h_{33}, \, h_{4}=2h_{44}, \\&h_{5}=2h_{55},\quad h_{6}=h_{66},\\&h_{7}=h_{77}, \quad h_{8}=h_{88},\quad h_{9}=2h_{99},\quad h_{10}=2h_{1010} \end{aligned}$$

$$\begin{aligned} l_{11}= & {} 1/4\,{\frac{B_{{14}}G_{{11}}-B_{{11}}G_{{13}}}{\omega _{{2}} \left( B_ {{12}}G_{{11}}-B_{{11}}G_{{12}} \right) }},\quad l_{22}=1/2\,{\frac{B_{{5}}}{B_{{8}}}},\\ l_{33}= & {} 1/2\,{\frac{B_{{6}}}{B_{{8}}}},\quad l_{44}={\frac{B_{{7}}}{B_{{8}}}}\\ l_{55}= & {} 1/4\,{\frac{B_{{13}}G_{{11}}-B_{{11}}G_{{14}}}{\omega _{{2}} \left( B_{{12}}G_{{11}}-B_{{11}}G_{{12}} \right) }}, \\ l_{66}= & {} 1/2\,{\frac{B_{{1}}}{B_{{4}}}},\quad l_{77}=1/2\,{\frac{B_{{2}}}{B_{{4}}}}, \\ l_{88}= & {} {\frac{B_{{3}}}{B_{{4}}}}, \\ l_{99}= & {} 1/16\,{\frac{G_{{11}}}{\omega _{{2}} \left( B_{{12}}G_{{11}}-B_{{11}}G _{{12}} \right) }}, \\ l_{1010}= & {} 1/16\,{\frac{B_{{11}}}{\omega _{{2}} \left( B_{{12}}G_{{11}}-B_{{11}}G _{{12}} \right) }} \\ l_{1}= & {} l_{11},\quad l_{2}=2l_{22},\,l_{3}=2l_{33}, \\ l_{4}= & {} 2l_{44},\quad l_{5}=2l_{55},\,l_{6}=l_{66},\\ l_{7}= & {} l_{77},\quad l_{8}=l_{88},\\ l_{9}= & {} 2l_{99},\quad l_{10}=2l_{1010} \end{aligned}$$