Nonlinear oscillation of active magnetic bearing–rotor systems with a time-delayed proportional–derivative controller

Abstract

A time-delayed proportional–derivative controller is applied to suppress the nonlinear vibration of a rigid rotor suspended by the active magnetic bearing (AMB) subjected to multiple excitations. The rotor eccentricity, the parametric excitations, the gravity of the rotor, and the coupling of the electromagnetic force between horizontal and vertical directions are taken into consideration in the dimensionless governing equation. The method of multiple scales is applied to obtain the approximate solution. Based on the Routh–Hurwitz stability criterion, the stability of steady-state solutions is discussed. Then, the influence of the control parameters and the time delay in the proportional and derivative control loop on the system is studied. And the optimal operating conditions are given. The results show that there exist multi-solutions and jump phenomena under certain conditions. Finally, the numerical solutions are obtained by applying ODE45 and DDE23 MATLAB solvers. And the analytical solutions have an excellent agreement with the numerical solutions.

Appendices

Appendix

Substituting Eq. (1), (2) and (4) into (3), the resultant force $F_x$ and $F_y$ can be obtained as:

$$\begin{aligned} F_x&=\left[ -I_b^2\mathrm{cos}^4\alpha -\left( I_b^2+i_0^2\right) \mathrm{sin}^4\alpha \right] \frac{16k}{g_0^5}x^3\nonumber \\&+\frac{24kI_bk_d\mathrm{cos}^3\alpha }{g_0^4}x^2{\dot{x}}_{\tau _2}+\frac{24kI_bk_p\mathrm{cos}^3\alpha }{g_0^4}x^2x_{\tau _1}\nonumber \\&-\left( 2I_b^2+i_0^2\right) \frac{48k\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha }{g_0^5}xy^2+\frac{48kI_bi_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}xy\nonumber \\&+\frac{48kk_pI_b\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}xyy_{\tau _1}+\frac{48kk_dI_b\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}xy{\dot{y}}_{\tau _2}\nonumber \\&-\frac{8kk_d^2\mathrm{cos}^2\alpha }{g_0^3}x{\dot{x}}^2_{\tau _2}-\frac{16kk_pk_d\mathrm{cos}^2\alpha }{g_0^3}xx_{\tau _1}{\dot{x}}_{\tau _2}-\frac{8kI_b^2\mathrm{cos}^2\alpha }{g_0^3}x\nonumber \\&-\frac{8k\left( I_b^2+i_0^2\right) \mathrm{sin}^2\alpha }{g_0^3}x-\frac{16kk_di_0\mathrm{sin}^2\alpha }{g_0^3}x{\dot{y}}_{\tau _2}\nonumber \\&-\frac{16kk_pi_0\mathrm{sin}^2\alpha }{g_0^3}xy_{\tau _1}-\frac{8kk_d^2\mathrm{sin}^2\alpha }{g_0^3}x{\dot{y}}^2_{\tau _2}-\frac{8kk_p^2\mathrm{sin}^2\alpha }{g_0^3}xy^2_{\tau _1}\nonumber \\&-\frac{16kk_pk_d\mathrm{sin}^2\alpha }{g_0^3}xy_{\tau _1}{\dot{y}}_{\tau _2}-\frac{8kk_p^2\mathrm{cos}^2\alpha }{g_0^3}xx^2_{\tau _1}\nonumber \\&+\frac{24kI_bk_d\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}y^2{\dot{x}}_{\tau _2}+\frac{24kI_bk_p\mathrm{sin}^2\mathrm{cos}\alpha }{g_0^4}y^2x_{\tau _1}\nonumber \\&+\frac{8kI_bk_d\mathrm{cos}\alpha }{g_0^2}{\dot{x}}_{\tau _2}+\frac{8kI_bk_p\mathrm{cos}\alpha }{g_0^2}x_{\tau _1} \end{aligned}$$

(A.1a)

$$\begin{aligned} F_y&=\frac{24kI_bi_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}x^2+\frac{24kk_pI_b\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}x^2y_{\tau _1}\nonumber \\&+\frac{24kk_dI_b\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}x^2{\dot{y}}_{\tau _2}-\frac{48k\left( I_b^2+i_0^2\right) \mathrm{sin}^2\alpha \mathrm{cos}^2\alpha }{g_0^5}x^2y\nonumber \\&-\frac{48kI_b^2\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha }{g_0^5}x^2y+\frac{48kI_bk_d\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}xy{\dot{x}}_{\tau _2}\nonumber \\&+\frac{48kI_bk_p\mathrm{sin}^2\alpha \mathrm{cos}\alpha }{g_0^4}xyx_{\tau _1}-\frac{16k\left( I_b^2+i_0^2\right) \mathrm{ocs}^4\alpha }{g_0^5}y^3\nonumber \\&-\frac{16kI_b^2\mathrm{sin}^4\alpha }{g_0^5}y^3+\frac{24kI_bi_0\mathrm{cos}^3\alpha }{g_0^4}y^2+\frac{24kk_pk_dI_b\mathrm{cos}^3\alpha }{g_0^4}y^2y_{\tau _1}\nonumber \\&+\frac{24kk_dI_b\mathrm{cos}^3\alpha }{g_0^4}y^2{\dot{y}}_{\tau _2}-\frac{8kk_d^2\mathrm{sin}^2\alpha }{g_0^3}y{\dot{x}}^2_{\tau _2}\nonumber \\&-\frac{16kk_pk_d\mathrm{sin}^2\alpha }{g_0^3}yx_{\tau _1}{\dot{x}}_{\tau _2}-\frac{8k\left( I_b^2+i^2_0\right) \mathrm{cos}^2\alpha }{g_0^3}y\nonumber \\&-\frac{8kI_b^2\mathrm{sin}^2\alpha }{g_0^3}y-\frac{16kk_pi_0\mathrm{cos}^2\alpha }{g_0^3}yy_{\tau _1}-\frac{16kk_di_0\mathrm{cos}^2\alpha }{g_0^3}y{\dot{y}}_{\tau _2}\nonumber \\&-\frac{8kk_d^2\mathrm{cos}^2\alpha }{g_0^3}y{\dot{y}}^2_{\tau _2}-\frac{8kk_p^2\mathrm{cos}^2\alpha }{g_0^3}yy^2_{\tau _1}-\frac{16kk_pk_d\mathrm{cos}^2\alpha }{g_0^3}yy_{\tau _1}{\dot{y}}_{\tau _2}\nonumber \\&-\frac{8kk_p^2\mathrm{sin}^2\alpha }{g_0^3}yx^2_{\tau _1}+\frac{8kI_bi_0\mathrm{cos}\alpha }{g_0^2}+\frac{8kk_dI_b\mathrm{cos}\alpha }{g_0^2}{\dot{y}}_{\tau _2}\nonumber \\&+\frac{8kk_pI_b\mathrm{cos}\alpha }{g_0^2}y_{\tau _1} \end{aligned}$$

(A.1b)

Substituting Eq. (A.1) into (5), it can be derived as:

$$\begin{aligned} \ddot{x}&+\mu _1{\dot{x}}-16[\mathrm{cos}^4\alpha +(1+i_0^2)\mathrm{sin}^4\alpha ]x^3+24d\mathrm{cos}^3\nonumber \\&\times \alpha x^2{\dot{x}}_{\tau _2}+24p\mathrm{cos}^3\alpha x^2x_{\tau _1}-48(2+i_0^2)\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha xy^2\nonumber \\&+48i_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha xy+48p\mathrm{sin}^2\alpha \mathrm{cos}\alpha xyy_{\tau _1}\nonumber \\&+48d\mathrm{sin}^2\alpha \mathrm{cos}\alpha xy{\dot{y}}_{\tau _2}-8d^2\mathrm{cos}^2\alpha x{\dot{x}}^2_{\tau _2}\nonumber \\&-16pd\mathrm{cos}^2\alpha xx_{\tau _1}{\dot{x}}_{\tau _2}-8\mathrm{cos}^2\alpha x-8(1+i_0^2)\mathrm{sin}^2\alpha x\nonumber \\&-16di_0\mathrm{sin}^2\alpha x{\dot{y}}_{\tau _2}-16pi_0\mathrm{sin}^2\alpha xy_{\tau _1}-8d^2\mathrm{sin}^2\alpha x {\dot{y}}^2_{\tau _2}\nonumber \\&-8p^2\mathrm{sin}^2\alpha xy^2_{\tau _1}-16pd\mathrm{sin}^2\alpha xy_{\tau _1}{\dot{y}}_{\tau _2}-8p^2\mathrm{cos}^2\alpha xx^2_{\tau _1}\nonumber \\&+24d\mathrm{sin}^2\alpha \mathrm{cos}\alpha y^2{\dot{x}}_{\tau _2}+24p\mathrm{sin}^2\alpha \mathrm{cos}\alpha y^2x_{\tau _1}\nonumber \\&+8d\mathrm{cos}\alpha {\dot{x}}_{\tau _2}+8p\mathrm{cos}\alpha x_{\tau _1}=\frac{2f\omega ^2}{\xi ^2}\mathrm{cos}(\omega t) \end{aligned}$$

(A.2a)

$$\begin{aligned} \ddot{y}&+\mu _1{\dot{y}}+24i_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha x^2+24p\mathrm{sin}^2\alpha \mathrm{cos}\alpha x^2y_{\tau _1}\nonumber \\&+24d\mathrm{sin}^2\alpha \mathrm{cos}\alpha x^2{\dot{y}}_{\tau _2}-48(1+i_0^2)\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha x^2y\nonumber \\&-48\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha x^2y+48d\mathrm{sin}^2\alpha \mathrm{cos}\alpha xy{\dot{x}}_{\tau _2}\nonumber \\&+48p\mathrm{sin}^2\alpha \mathrm{cos}\alpha xyx_{\tau _1}-16(1+i_0^2)\mathrm{cos}^4\alpha y^3 \nonumber \\&-16\mathrm{sin}^4\alpha y^3+24i_0\mathrm{cos}^3\alpha y^2+24p\mathrm{cos}^3\alpha y^2y_{\tau _1}\nonumber \\&+24d\mathrm{cos}^3\alpha y^2{\dot{y}}_{\tau _2}-8d^2\mathrm{sin}^2\alpha y{\dot{x}}^2_{\tau _2}\nonumber \\&-16pd\mathrm{sin}^2\alpha yx_{\tau _1}{\dot{x}}_{\tau _2}-8(1+i_0^2)\mathrm{cos}^2\alpha y\nonumber \\&-8\mathrm{sin}^2\alpha y-16pi_0\mathrm{cos}^2\alpha yy_{\tau _1}-16di_0\mathrm{cos}^2\alpha y{\dot{y}}_{\tau _2}\nonumber \\&-8d^2\mathrm{cos}^2\alpha y{\dot{y}}^2_{\tau _2}-8p^2\mathrm{cos}^2\alpha yy^2_{\tau _1}-16pd\mathrm{cos}^2\nonumber \\&\alpha yy_{\tau _1}{\dot{y}}_{\tau _2}-8p^2\mathrm{sin}^2\alpha yx^2_{\tau _1}+8d^2\mathrm{cos}\alpha {\dot{y}}_{\tau _2}+8p\mathrm{cos}\alpha y_{\tau _1}\nonumber \\&=\frac{2f\omega ^2}{\xi ^2}\mathrm{sin}(\omega t) \end{aligned}$$

(A.2b)

where $p=\frac{kI_bk_p}{\xi ^2mg_0^2}$, $d=\frac{kk_dI_b}{\xi mg_0^2}$, $\mu _1=\frac{c}{m\xi }$, $f=\frac{m_ue}{2mg_0}$, $\xi =\sqrt{\frac{kI_b^2}{mg_0^3}}$. Introducing non-dimensional parameters $t^*=\xi t,\omega =\xi \varOmega ,x=g_0x^*,y=g_0y^*,i_0=I_bi_0^*$, and omitting the asterisk for brevity, Eq. (A.2) can be rewritten as:

$$\begin{aligned} \ddot{x}&+\mu _1{\dot{x}}+\omega _1^2x+\gamma _1xy+\gamma _2xy^2+\gamma _3x^3=\alpha _1xy_{\tau _1}\nonumber \\&+\alpha _2x{\dot{y}}_{\tau _2}+\alpha _3\left( x-x_{\tau _1}\right) +\alpha _4{\dot{x}}_{\tau _2}+\alpha _5x^2{\dot{x}}_{\tau _2}\nonumber \\&+\alpha _6x^2x_{\tau _1}+\alpha _7xyy_{\tau _1}+\alpha _8xy{\dot{y}}_{\tau _2}+\alpha _9x{\dot{x}}^2_{\tau _2}\nonumber \\&+\alpha _{10}xx_{\tau _1}{\dot{x}}_{\tau _2}+\alpha _{11}x{\dot{y}}^2_{\tau _2}+\alpha _{12}xy^2_{\tau _1}\nonumber \\&+\alpha _{13}xy_{\tau _1}{\dot{y}}_{\tau _2}+\alpha _{14}xx^2_{\tau _1}+\alpha _{15}y^2{\dot{x}}_{\tau _2}+\alpha _{16}y^2x_{\tau _1}\nonumber \\&+2f\varOmega ^2\mathrm{cos}\left( \varOmega t\right) \end{aligned}$$

(A.3a)

$$\begin{aligned} \ddot{y}&+\mu _1{\dot{y}}+\omega ^2_2y+\gamma _4x^2+\gamma _5y^2+\gamma _6x^2y+\gamma _7y^3\nonumber \\&=\beta _1yy_{\tau _1}+\beta _2y{\dot{y}}_{\tau _2}+\beta _3\left( y-y_{\tau _1}\right) +\beta _4{\dot{y}}_{\tau _2}\nonumber \\&+\beta _5x^2y_{\tau _1}+\beta _6x^2{\dot{y}}_{\tau _2}+\beta _7xy{\dot{x}}_{\tau _2}+\beta _8xyx_{\tau _1}\nonumber \\&+\beta _9y^2y_{\tau _1}+\beta _{10}y^2{\dot{y}}_{\tau _2}+\beta _{11}y{\dot{x}}^2_{\tau _2}+\beta _{12}yx_{\tau _1}{\dot{x}}_{\tau _2}\nonumber \\&+\beta _{13}y{\dot{y}}^2_{\tau _2}+\beta _{14}yy^2_{\tau _1}+\beta _{15}yy_{\tau _1}{\dot{y}}_{\tau _2}+\beta _{16}yx^2_{\tau _1}\nonumber \\&+2f\varOmega \mathrm{sin}\left( \varOmega t\right) \end{aligned}$$

(A.3b)

where the coefficients are listed in the following equations:

$$\begin{aligned} \omega _1= & {} \sqrt{8(p\mathrm{cos}\alpha -1)-8i_0^2\mathrm{sin}^2\alpha } \end{aligned}$$

(A.4)

$$\begin{aligned} \omega _2= & {} \sqrt{8(p\mathrm{cos}\alpha -1)-8i_0^2\mathrm{cos}^2\alpha } \end{aligned}$$

(A.5)

$$\begin{aligned} \gamma _1= & {} 48i_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.6)

$$\begin{aligned} \gamma _2= & {} -48(2+i_0^2)\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha \end{aligned}$$

(A.7)

$$\begin{aligned} \gamma _3= & {} -16\mathrm{cos}^4\alpha -16(1+i_0^2)\mathrm{sin}^4\alpha \end{aligned}$$

(A.8)

$$\begin{aligned} \gamma _4= & {} 24i_0\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.9)

$$\begin{aligned} \gamma _5= & {} 24i_0\mathrm{cos}^3\alpha \end{aligned}$$

(A.10)

$$\begin{aligned} \gamma _6= & {} -48(2+i_0^2)\mathrm{sin}^2\alpha \mathrm{cos}^2\alpha \end{aligned}$$

(A.11)

$$\begin{aligned} \gamma _7= & {} -16(1+i_0^2)\mathrm{cos}^4\alpha -16\mathrm{sin}^4\alpha \end{aligned}$$

(A.12)

$$\begin{aligned} \alpha _1= & {} 16pi_0\mathrm{sin}^2\alpha \end{aligned}$$

(A.13)

$$\begin{aligned} \alpha _2= & {} 16di_0\mathrm{sin}^2\alpha \end{aligned}$$

(A.14)

$$\begin{aligned} \alpha _3= & {} 8p\mathrm{cos}\alpha \end{aligned}$$

(A.15)

$$\begin{aligned} \alpha _4= & {} -8d\mathrm{cos}\alpha \end{aligned}$$

(A.16)

$$\begin{aligned} \alpha _5= & {} -24d\mathrm{cos}^3\alpha \end{aligned}$$

(A.17)

$$\begin{aligned} \alpha _6= & {} -24p\mathrm{cos}^3\alpha \end{aligned}$$

(A.18)

$$\begin{aligned} \alpha _7= & {} -48p\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.19)

$$\begin{aligned} \alpha _8= & {} -48d\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.20)

$$\begin{aligned} \alpha _9= & {} 8d^2\mathrm{cos}^2\alpha \end{aligned}$$

(A.21)

$$\begin{aligned} \alpha _{10}= & {} 16pd\mathrm{cos}^2\alpha \end{aligned}$$

(A.22)

$$\begin{aligned} \alpha _{11}= & {} 8d^2\mathrm{sin}^2\alpha \end{aligned}$$

(A.23)

$$\begin{aligned} \alpha _{12}= & {} 8p^2\mathrm{sin}^2\alpha \end{aligned}$$

(A.24)

$$\begin{aligned} \alpha _{13}= & {} 16pd\mathrm{sin}^2\alpha \end{aligned}$$

(A.25)

$$\begin{aligned} \alpha _{14}= & {} 8p^2\mathrm{cos}^2\alpha \end{aligned}$$

(A.26)

$$\begin{aligned} \alpha _{15}= & {} -24d\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.27)

$$\begin{aligned} \alpha _{16}= & {} -24p\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.28)

$$\begin{aligned} \beta _1= & {} 16pi_0\mathrm{cos}^2\alpha \end{aligned}$$

(A.29)

$$\begin{aligned} \beta _2= & {} 16di_0\mathrm{cos}^2\alpha \end{aligned}$$

(A.30)

$$\begin{aligned} \beta _3= & {} 8p\mathrm{cos}\alpha \end{aligned}$$

(A.31)

$$\begin{aligned} \beta _4= & {} -8d\mathrm{cos}\alpha \end{aligned}$$

(A.32)

$$\begin{aligned} \beta _5= & {} -24p\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.33)

$$\begin{aligned} \beta _6= & {} -24d\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.34)

$$\begin{aligned} \beta _7= & {} -48d\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.35)

$$\begin{aligned} \beta _8= & {} -48p\mathrm{sin}^2\alpha \mathrm{cos}\alpha \end{aligned}$$

(A.36)

$$\begin{aligned} \beta _9= & {} -24p\mathrm{cos}^3\alpha \end{aligned}$$

(A.37)

$$\begin{aligned} \beta _{10}= & {} -24d\mathrm{cos}^3\alpha \end{aligned}$$

(A.38)

$$\begin{aligned} \beta _{11}= & {} 8d^2\mathrm{sin}^2\alpha \end{aligned}$$

(A.39)

$$\begin{aligned} \beta _{12}= & {} 16pd\mathrm{sin}^2\alpha \end{aligned}$$

(A.40)

$$\begin{aligned} \beta _{13}= & {} 8d^2\mathrm{cos}^2\alpha \end{aligned}$$

(A.41)

$$\begin{aligned} \beta _{14}= & {} 8p^2\mathrm{cos}^2\alpha \end{aligned}$$

(A.42)

$$\begin{aligned} \beta _{15}= & {} 16pd\mathrm{cos}^2\alpha \end{aligned}$$

(A.43)

$$\begin{aligned} \beta _{16}= & {} 8p^2\mathrm{sin}^2\alpha \end{aligned}$$

(A.44)

Appendix

$$\begin{aligned} \varGamma _{11}&= \mathrm{j}\omega _1{\hat{\mu }}_1-{\hat{\alpha }}_3+{\hat{\alpha }}_3\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}-\mathrm{j}\omega _1{\hat{\alpha }}_4\mathrm{e}^{-\mathrm{j}\omega _1\tau _2} \end{aligned}$$

(B.1)

$$\begin{aligned} \varGamma _{12}=&\,\frac{2{\hat{\gamma }}_1{\hat{\gamma }}_4}{\omega _2^2}-\frac{{\hat{\gamma }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}-3{\hat{\gamma }}_3-\frac{2{\hat{\alpha }}_1{\hat{\gamma }}_4}{\omega _2^2}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1}+\frac{2\mathrm{j}\omega _1{\hat{\alpha }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&-\mathrm{j}\omega _1{\hat{\alpha }}_5\mathrm{e}^{\mathrm{j}\omega _1\tau _2}+2\mathrm{j}\omega _1{\hat{\alpha }}_5\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}+{\hat{\alpha }}_6\mathrm{e}^{\mathrm{j}\omega _1\tau _1}\nonumber \\&+2{\hat{\alpha }}_6\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}+2{\hat{\alpha }}_9\omega _1^2-{\hat{\alpha }}_9\omega _1^2\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&-\mathrm{j}\omega _1{\hat{\alpha }}_{10}\mathrm{e}^{-\mathrm{j}\omega _1\left( \tau _1-\tau _2\right) }+\mathrm{j}\omega _1{\hat{\alpha }}_{10}\mathrm{e}^{\mathrm{j}\omega _1\left( \tau _1-\tau _2\right) }\nonumber \\&+\mathrm{j}\omega _1{\hat{\alpha }}_{10}\mathrm{e}^{-\mathrm{j}\omega _1\left( \tau _1+\tau _2\right) }+2{\hat{\alpha }}_{14}+{\hat{\alpha }}_{14}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.2)

$$\begin{aligned} \varGamma _{13}=&\frac{2{\hat{\gamma }}_1{\hat{\gamma }}_5-2{\hat{\gamma }}_1{\hat{\beta }}_1\mathrm{cos}\left( \omega _2\tau _1\right) -2\omega _2{\hat{\gamma }}_1{\hat{\beta }}_2\mathrm{sin}\left( \omega _2\tau _2\right) }{\omega _2^2}\nonumber \\&-\frac{{\hat{\gamma }}_1^2-{\hat{\gamma }}_1{\hat{\alpha }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }-2{\hat{\gamma }}_2\nonumber \\&-\frac{{\hat{\gamma }}_1^2-{\hat{\gamma }}_1{\hat{\alpha }}_1\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\alpha }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }+{\hat{\alpha }}_7\mathrm{e}^{\mathrm{j}\omega _2\tau _1}\nonumber \\&-\frac{2{\hat{\alpha }}_1{\hat{\gamma }}_5-2{\hat{\alpha }}_1{\hat{\beta }}_1\mathrm{cos}\left( \omega _2\tau _1\right) -2\omega _2{\hat{\alpha }}_1{\hat{\beta }}_2\mathrm{sin}\left( \omega _2\tau _2\right) }{\omega _2^2}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_1-{\hat{\alpha }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }\mathrm{e}^{\mathrm{j}\omega _2\tau _1}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_1-{\hat{\alpha }}_1^2\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&-\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\gamma }}_1-{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }\mathrm{e}^{\mathrm{j}\omega _2\tau _2}\nonumber \\&+\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\gamma }}_1-{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\alpha }}_2^2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+{\hat{\alpha }}_7\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_8\mathrm{e}^{\mathrm{j}\omega _2\tau _2}+\mathrm{j}\omega _2{\hat{\alpha }}_8\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+2{\hat{\alpha }}_{11}\omega _2^2+2{\hat{\alpha }}_{12}-\mathrm{j}\omega _2{\hat{\alpha }}_{13}\mathrm{e}^{\mathrm{j}\omega _2\left( \tau _2-\tau _1\right) }+2{\hat{\alpha }}_{16}\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}\nonumber \\&+\mathrm{j}\omega _2{\hat{\alpha }}_{13}\mathrm{e}^{\mathrm{j}\omega _2\left( \tau _1-\tau _2\right) }+2\mathrm{j}\omega _1{\hat{\alpha }}_{15}\mathrm{e}^{-\mathrm{j}\omega _1\tau _2} \end{aligned}$$

(B.3)

$$\begin{aligned} \varGamma _{14}=&-\frac{{\hat{\gamma }}_1{\hat{\gamma }}_5-{\hat{\gamma }}_1{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}+{\hat{\alpha }}_7\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&-\frac{{\hat{\gamma }}_1^2-{\hat{\gamma }}_1{\hat{\alpha }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }+\mathrm{j}\omega _2{\hat{\alpha }}_8\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_5-{\hat{\alpha }}_1{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}-{\hat{\gamma }}_2\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_1-{\hat{\alpha }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&+2\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\alpha }}_5-{\hat{\alpha }}_2{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_2{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}\nonumber \\&+\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\gamma }}_1-{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&-{\hat{\alpha }}_{11}\omega _2^2\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}+{\hat{\alpha }}_{12}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\alpha }}_{13}\mathrm{e}^{-\mathrm{j}\omega _2\left( \tau _1+\tau _2\right) }\nonumber \\&+\mathrm{j}\omega _1{\hat{\alpha }}_{15}\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}+{\hat{\alpha }}_{16}\mathrm{e}^{-\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.4)

$$\begin{aligned} \varGamma _{15}=&-\frac{{\hat{\gamma }}_1{\hat{\gamma }}_5-{\hat{\gamma }}_1{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\nonumber \\&+{\hat{\alpha }}_7\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&-\frac{{\hat{\gamma }}_1^2-{\hat{\gamma }}_1{\hat{\alpha }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\nonumber \\&+\mathrm{j}\omega _2{\hat{\alpha }}_8\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_5-{\hat{\alpha }}_1{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\nonumber \\&\times \mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}\nonumber \\&+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_1-{\hat{\alpha }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\nonumber \\&\times \mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-{\hat{\gamma }}_2\nonumber \\&+2\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\gamma }}_5-{\hat{\alpha }}_2{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_2{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\nonumber \\&\times \mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}\nonumber \\&+\mathrm{j}\omega _2\frac{{\hat{\alpha }}_2{\hat{\gamma }}_1-{\hat{\alpha }}_1{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\alpha }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\nonumber \\&\times \mathrm{e}^{-\mathrm{j}\omega _2\tau _2}-{\hat{\alpha }}_{11}\omega _2^2\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}+{\hat{\alpha }}_{12}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}\nonumber \\&+\mathrm{j}\omega _2{\hat{\alpha }}_{13}\mathrm{e}^{-\mathrm{j}\omega _2\left( \tau _1+\tau _2\right) }-\mathrm{j}\omega _1{\hat{\alpha }}_{15}\mathrm{e}^{\mathrm{j}\omega _1\tau _2}\nonumber \\&+{\hat{\alpha }}_{16}\mathrm{e}^{\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.5)

$$\begin{aligned} \varGamma _{16}=&-\frac{{\hat{\gamma }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}-{\hat{\gamma }}_3+\frac{{\hat{\alpha }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1}\nonumber \\&+\mathrm{j}\omega _1{\hat{\alpha }}_5\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}+2\mathrm{j}\omega _1\frac{{\hat{\alpha }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&+{\hat{\alpha }}_6\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}-{\hat{\alpha }}_9\omega _1^2\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&+\mathrm{j}\omega _1{\hat{\alpha }}_{10}\mathrm{e}^{-\mathrm{j}\omega _1\left( \tau _1+\tau _2\right) }+{\hat{\alpha }}_{14}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.6)

$$\begin{aligned} \varGamma _{21}=&\mathrm{j}\omega _2{\hat{\mu }}_1-\mathrm{j}\omega _2{\hat{\beta }}_4\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}+{\hat{\beta }}_3\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-{\hat{\beta }}_3 \end{aligned}$$

(B.7)

$$\begin{aligned} \varGamma _{22}=&-4\frac{{\hat{\gamma }}_1{\hat{\gamma }}_4-{\hat{\gamma }}_4{\hat{\alpha }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_4{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2^2-4\omega _1^2}+\frac{4{\hat{\gamma }}_4{\hat{\gamma }}_5}{\omega _2^2}\nonumber \\&-2{\hat{\gamma }}_6-\frac{2{\hat{\beta }}_1{\hat{\gamma }}_4}{\omega _2^2}-\frac{2{\hat{\beta }}_1{\hat{\gamma }}_4}{\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-2\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_4}{\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+2{\hat{\beta }}_5\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}+2\mathrm{j}\omega _2{\hat{\beta }}_6\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}-\mathrm{j}\omega _1{\hat{\beta }}_7\mathrm{e}^{\mathrm{j}\omega _1\tau _2}\nonumber \\&+\mathrm{j}\omega _1{\hat{\beta }}_7\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}+{\hat{\beta }}_8\mathrm{e}^{\mathrm{j}\omega _1\tau _1}+{\hat{\beta }}_8\mathrm{e}^{{-\mathrm j}\omega _1\tau _1}+2{\hat{\beta }}_{11}\omega _1^2\nonumber \\&-\mathrm{j}\omega _1{\hat{\beta }}_{12}\mathrm{e}^{\mathrm{j}\omega _1\left( \tau _2-\tau _1\right) }+\mathrm{j}\omega _1{\hat{\beta }}_{12}\mathrm{e}^{\mathrm{j}\omega _1\left( \tau _1-\tau _2\right) }+2{\hat{\beta }}_{16} \end{aligned}$$

(B.8)

$$\begin{aligned} \varGamma _{23}=&4\frac{{\hat{\gamma }}_5^2-{\hat{\gamma }}_5{\hat{\beta }}_1\mathrm{cos}\left( \omega _2\tau _1\right) -\omega _2{\hat{\gamma }}_5{\hat{\beta }}_2\mathrm{sin}\left( \omega _2\tau _2\right) }{\omega _2^2}-3{\hat{\gamma }}_7\nonumber \\&-2\frac{{\hat{\gamma }}_5^2-{\hat{\gamma }}_5{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_5{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}+{\hat{\beta }}_9\mathrm{e}^{\mathrm{j}\omega _2\tau _1}\nonumber \\&-2\frac{{\hat{\beta }}_1{\hat{\beta }}_5-{\hat{\beta }}_1^2\mathrm{cos}\left( \omega _2\tau _1\right) -\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{sin}\left( \omega _2\tau _2\right) }{\omega _2^2}\nonumber \\&+\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}\nonumber \\&+\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{\mathrm{j}\omega _2\tau _1}\nonumber \\&-\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&-\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2^2}\nonumber \\&\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}+2{\hat{\beta }}_9\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&-2\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}\nonumber \\&-\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{\mathrm{j}\omega _2\tau _2}\nonumber \\&-\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&-\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2^2}\nonumber \\&\times \mathrm{e}^{-\mathrm{j}\omega _2\tau _2}+2\omega _2^2{\hat{\beta }}_{13}\nonumber \\&-\mathrm{j}\omega _2{\hat{\beta }}_{10}\mathrm{e}^{\mathrm{j}\omega _2\tau _2}+2\mathrm{j}\omega _2{\hat{\beta }}_{10}\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}-\omega _2^2{\hat{\beta }}_{13}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}\nonumber \\&+2{\hat{\beta }}_{14}+{\hat{\beta }}_{14}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_{15}\mathrm{e}^{\mathrm{j}\omega _2\left( \tau _2-\tau _1\right) }\nonumber \\&+\mathrm{j}\omega _2{\hat{\beta }}_{15}\mathrm{e}^{\mathrm{j}\omega _2\left( \tau _1-\tau _2\right) }+\mathrm{j}\omega _2{\hat{\beta }}_{15}\mathrm{e}^{-\mathrm{j}\omega _2\left( \tau _2+\tau _1\right) } \end{aligned}$$

(B.9)

$$\begin{aligned} \varGamma _{24}=&-2\frac{{\hat{\gamma }}_1{\hat{\gamma }}_4-{\hat{\gamma }}_4{\hat{\alpha }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_4{\hat{\alpha }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{\omega _2\left( 2\omega _1+\omega _2\right) }\nonumber \\&-\frac{2{\hat{\gamma }}_4{\hat{\gamma }}_5}{4\omega _1^2-\omega _2^2}+\frac{{\hat{\beta }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1}+\frac{{\hat{\beta }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\nonumber \\&\times \mathrm{e}^{-\mathrm{j}\omega _2\tau _1}+{\hat{\beta }}_5\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&+2\mathrm{j}\omega _1\frac{{\hat{\beta }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}+\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\nonumber \\&\times \mathrm{e}^{-\mathrm{j}\omega _2\tau _2}-{\hat{\gamma }}_6+\mathrm{j}\omega _2{\hat{\beta }}_6\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}+\mathrm{j}\omega _1{\hat{\beta }}_7\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}\nonumber \\&+\mathrm{j}\omega _1{\hat{\beta }}_{12}\mathrm{e}^{-\mathrm{j}\omega _1\left( \tau _1+\tau _2\right) }+{\hat{\beta }}_8\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}-{\hat{\beta }}_{11}\omega _1^2\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&+{\hat{\beta }}_{16}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.10)

$$\begin{aligned} \varGamma _{25}=&-2\frac{{\hat{\gamma }}_1{\hat{\gamma }}_4-{\hat{\gamma }}_4{\hat{\alpha }}_1\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\mathrm{j}\omega _2{\hat{\gamma }}_4{\hat{\alpha }}_2\mathrm{e}^{\mathrm{j}\omega _2\tau _2}}{\omega _2\left( \omega _2-2\omega _1\right) }\nonumber \\&+\frac{{\hat{\beta }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{\mathrm{j}\omega _2\tau _1}+\frac{{\hat{\beta }}_1{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1}\nonumber \\&-\frac{2{\hat{\gamma }}_4{\hat{\gamma }}_5}{4\omega _1^2-\omega _2^2}-\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{\mathrm{j}\omega _2\tau _2}\nonumber \\&+2\mathrm{j}\omega _1\frac{{\hat{\beta }}_2{\hat{\gamma }}_4}{4\omega _1^2-\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}-{\hat{\gamma }}_6+{\hat{\beta }}_5\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&+\mathrm{j}\omega _2{\hat{\beta }}_6\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}+\mathrm{j}\omega _1{\hat{\beta }}_7\mathrm{e}^{-\mathrm{j}\omega _1\tau _2}\nonumber \\&+{\hat{\beta }}_8\mathrm{e}^{-\mathrm{j}\omega _1\tau _1}-{\hat{\beta }}_{11}\omega _1^2\mathrm{e}^{-2\mathrm{j}\omega _1\tau _2}\nonumber \\&+\mathrm{j}\omega _1{\hat{\beta }}_{12}\mathrm{e}^{-\mathrm{j}\omega _1\left( \tau _1+\tau _2\right) }+{\hat{\beta }}_{16}\mathrm{e}^{-2\mathrm{j}\omega _1\tau _1} \end{aligned}$$

(B.11)

$$\begin{aligned} \varGamma _{26}=&-2\frac{{\hat{\gamma }}_5^2-{\hat{\gamma }}_5{\hat{\beta }}_1\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\gamma }}_5{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\nonumber \\&-{\hat{\gamma }}_7+{\hat{\beta }}_9\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&+\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}\nonumber \\&+\frac{{\hat{\beta }}_1{\hat{\gamma }}_5-{\hat{\beta }}_1^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}\nonumber \\&+2\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}\nonumber \\&+\mathrm{j}\omega _2\frac{{\hat{\beta }}_2{\hat{\gamma }}_5-{\hat{\beta }}_1{\hat{\beta }}_2\mathrm{e}^{-\mathrm{j}\omega _2\tau _1}-\mathrm{j}\omega _2{\hat{\beta }}_2^2\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}}{3\omega _2^2}\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}\nonumber \\&+\mathrm{j}\omega _2{\hat{\beta }}_{10}\mathrm{e}^{-\mathrm{j}\omega _2\tau _2}-\omega _2^2{\hat{\beta }}_{13}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _2}+{\hat{\beta }}_{14}\mathrm{e}^{-2\mathrm{j}\omega _2\tau _1}\nonumber \\&+\mathrm{j}\omega _2{\hat{\beta }}_{15}\mathrm{e}^{-\mathrm{j}\omega _2\left( \tau _1+\tau _2\right) } \end{aligned}$$

(B.12)

Appendix

$$\begin{aligned} \varLambda _1=&-\omega _1\mu _1+\alpha _3\mathrm{sin}\left( \omega _1\tau _1\right) +\omega _1\alpha _4\mathrm{cos}\left( \omega _1\tau _2\right) \end{aligned}$$

(C.1)

$$\begin{aligned} \varXi _1=&-\alpha _3\mathrm{cos}\left( \omega _1\tau _1\right) +\alpha _3+\omega _1\alpha _4\mathrm{sin}\left( \omega _1\tau _2\right) \end{aligned}$$

(C.2)

$$\begin{aligned} \varLambda _2=&-\frac{\alpha _1\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( 2\omega _1\tau _1\right) +\frac{2\omega _1\alpha _2\gamma _4}{4\omega _1^2-\omega _2^2}\nonumber \\&\times \mathrm{cos}\left( 2\omega _1\tau _2\right) +\omega _1\alpha _5\mathrm{cos}\left( \omega _1\tau _2\right) -\alpha _6\mathrm{sin}\left( \omega _1\tau _1\right) \nonumber \\&+\alpha _9\omega _1^2\mathrm{sin}\left( 2\omega _1\tau _2\right) +\omega _1\alpha _{10}\mathrm{cos}\left( \omega _1\tau _1+\omega _1\tau _2\right) \nonumber \\&-\alpha _{14}\mathrm{sin}\left( 2\omega _1\tau _1\right) \end{aligned}$$

(C.3)

$$\begin{aligned} \varXi _2=&-\frac{\gamma _1\gamma _4}{4\omega _1^2-\omega _2^2}+\frac{2\gamma _1\gamma _4}{\omega _2^2}-3\gamma _3+\frac{\alpha _1\gamma _4}{4\omega _1^2-\omega _2^2}\nonumber \\&\times \mathrm{cos}\left( 2\omega _1\tau _1\right) -\frac{2\alpha _1\gamma _4}{\omega _2^2}+\frac{2\omega _1\alpha _2\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( 2\omega _1\tau _2\right) \nonumber \\&+3\omega _1\alpha _5\mathrm{sin}\left( \omega _1\tau _2\right) +3\alpha _6\mathrm{cos}\left( \omega _1\tau _1\right) +2\alpha _9\omega _1^2\nonumber \\&-\alpha _9\omega _1^2\mathrm{cos}\left( 2\omega _1\tau _2\right) +2\alpha _{14}\nonumber \\&-2\omega _1\alpha _{10}\mathrm{sin}\left( \omega _1\tau _1-\omega _1\tau _2\right) \nonumber \\&+\omega _1\alpha _{10}\mathrm{sin}\left( \omega _1\tau _1+\omega _1\tau _2\right) +\alpha _{14}\mathrm{cos}\left( 2\omega _1\tau _1\right) \end{aligned}$$

(C.4)

$$\begin{aligned} \varLambda _3=&\alpha _{15}\omega _1\mathrm{cos}\left( \omega _1\tau _2\right) -\alpha _{16}\mathrm{sin}\left( \omega _1\tau _1\right) \end{aligned}$$

(C.5)

$$\begin{aligned} \varXi _3=&\frac{\gamma _1\gamma _5}{\omega _2^2}-\frac{\gamma _1\beta _1}{\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) -\frac{\gamma _1\beta _2}{\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\frac{\gamma _1^2}{4\omega _1^2-\omega _2^2}-\frac{2\gamma _1\alpha _1}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) \nonumber \\&-\frac{2\omega _2\gamma _1\alpha _2}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _2\right) -\frac{\alpha _1\gamma _5}{\omega _2^2}-\gamma _2\nonumber \\&+\frac{\alpha _1\beta _1}{\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) +\frac{\alpha _1\beta _2}{\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\frac{\alpha _1^2}{4\omega _1^2-\omega _2^2+\frac{2\omega _2\alpha _1\alpha _2}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _2-\omega _2\tau _1\right) }\nonumber \\&+\frac{\omega _2^2\alpha _2^2}{4\omega _1^2-\omega _2^2}+\alpha _{11}\omega _2^2+\alpha _7\mathrm{cos}\left( \omega _2\tau _1\right) \nonumber \\&+\alpha _8\omega _2\mathrm{sin}\left( \omega _2\tau _2\right) +\alpha _{15}\omega _1\mathrm{sin}\left( \omega _1\tau _2\right) +\alpha _{12}\nonumber \\&-\alpha _{13}\omega _2\mathrm{sin}\left( \omega _2\tau _1-\omega _2\tau _2\right) +\alpha _{16}\mathrm{cos}\left( \omega _1\tau _1\right) \end{aligned}$$

(C.6)

$$\begin{aligned} \varLambda _4=&-\frac{\gamma _1\beta _1}{3\omega _2^2}\mathrm{sin}\left( \omega _2\tau _1\right) +\frac{\gamma _1\beta _2}{3\omega _2}\left( \omega _2\tau _2\right) -\frac{\alpha _1\gamma _5}{3\omega _2^2}\mathrm{sin}\left( 2\omega _2\tau _1\right) \nonumber \\&-\frac{2\gamma _1\alpha _1}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{sin}\left( \omega _2\tau _1\right) -\frac{\alpha _1\beta _2}{3\omega _2}\mathrm{cos}\left( 2\omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&+\frac{\alpha _1\beta _1}{3\omega _2^2}\mathrm{sin}\left( 3\omega _2\tau _1\right) -\frac{2\alpha _1\alpha _2}{\omega _2-2\omega _1}\mathrm{cos}\left( \omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&+\frac{\alpha _1^2}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{sin}\left( 2\omega _2\tau _1\right) -\frac{\omega _2\alpha _2^2}{\omega _2-2\omega _1}\mathrm{sin}\left( 2\omega _2\tau _2\right) \nonumber \\&-\frac{2\alpha _2\beta _1}{3\omega _2}\mathrm{cos}\left( \omega _2\tau _1+2\omega _2\tau _1\right) +\frac{2\alpha _2\gamma _5}{3\omega _2}\mathrm{cos}\left( 2\omega _2\tau _2\right) \nonumber \\&-\frac{2\alpha _2\beta _2}{3}\mathrm{sin}\left( 3\omega _2\tau _2\right) -\alpha _7\mathrm{sin}\left( \omega _2\tau _1\right) -\alpha _{15}\omega _1\mathrm{cos}\left( \omega _1\tau _2\right) \nonumber \\&+\alpha _{11}\omega _2^2\mathrm{sin}\left( 2\omega _2\tau _2\right) +\alpha _8\omega _2\mathrm{cos}\left( \omega _2\tau _2\right) -\alpha _{12}\mathrm{sin}\left( 2\omega _2\tau _1\right) \nonumber \\&+\alpha _{13}\omega _2\mathrm{cos}\left( \omega _2\tau _1+\omega _2\tau _2\right) +\alpha _{16}\mathrm{sin}\left( \omega _1\tau _1\right) \end{aligned}$$

(C.7)

$$\begin{aligned} \varXi _4=&+\frac{\gamma _1\beta _1}{3\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) +\frac{\gamma _1\beta _2}{3\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) +\frac{\gamma _1^2}{\omega _2\left( 2\omega _1-\omega _2\right) }\nonumber \\&-\gamma _2+\frac{2\gamma _1\alpha _1}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{cos}\left( \omega _2\tau _1\right) +\frac{2\gamma _1\alpha _2}{\omega _2-2\omega _1}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&-\frac{\gamma _1\gamma _5}{3\omega _2^2}+\frac{\alpha _1\gamma _5}{3\omega _2^2}\mathrm{cos}\left( 2\omega _2\tau _1\right) -\frac{\alpha _1\beta _2}{3\omega _2}\mathrm{sin}\left( 2\omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&-\frac{\alpha _1\beta _1}{3\omega _2^2}\mathrm{cos}\left( 3\omega _2\tau _1\right) -\frac{2\alpha _1\alpha _2}{\omega _2-2\omega _1}\mathrm{sin}\left( \omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&-\frac{\alpha _1^2}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{cos}\left( 2\omega _2\tau _1\right) +\frac{2\alpha _2\gamma _5}{3\omega _2}\mathrm{sin}\left( 2\omega _2\tau _2\right) \nonumber \\&-\frac{2\alpha _2\beta _1}{3\omega _2}\mathrm{sin}\left( \omega _2\tau _1+2\omega _2\tau _2\right) +\frac{\omega _2\alpha _2^2}{\omega _2-2\omega _1}\mathrm{cos}\left( 2\omega _2\tau _2\right) \nonumber \\&+\frac{2\alpha _2\beta _2}{3}\mathrm{cos}\left( 3\omega _2\tau _2\right) +\alpha _7\mathrm{cos}\left( \omega _2\tau _1\right) +\alpha _8\omega _2\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&-\alpha _{11}\omega _2^2\mathrm{cos}\left( 2\omega _2\tau _2\right) +\alpha _{12}\mathrm{cos}\left( 2\omega _2\tau _1\right) +\alpha _{16}\mathrm{cos}\left( \omega _1\tau _1\right) \nonumber \\&+\alpha _{13}\omega _2\mathrm{cos}\left( \omega _2\tau _1+\omega _2\tau _2\right) +\alpha _{15}\omega _1\mathrm{sin}\left( \omega _1\tau _2\right) \end{aligned}$$

(C.8)

$$\begin{aligned} \varLambda _5=&-\omega _2\mu _1+\beta _4\omega _2\mathrm{cos}\left( \omega _2\tau _2\right) +\beta _3\mathrm{sin}\left( \omega _2\tau _1\right) \end{aligned}$$

(C.9)

$$\begin{aligned} \varXi _5=&\beta _4\omega _2\mathrm{sin}\left( \omega _2\tau _2\right) -\beta _3\mathrm{cos}\left( \omega _2\tau _1\right) +\beta _3 \end{aligned}$$

(C.10)

$$\begin{aligned} \varLambda _6=&\frac{5\gamma _5\beta _1}{3\omega _2^2}\mathrm{sin}\left( \omega _2\tau _1\right) -\frac{5\gamma _5\beta _2}{3\omega _2}\mathrm{cos}\left( \omega _2\tau _2\right) -\frac{\beta _1\gamma _5}{3\omega _2^2}\mathrm{sin}\left( 2\omega _2\tau _1\right) \nonumber \\&-\frac{\beta _1\beta _2}{3\omega _2}\mathrm{cos}\left( 2\omega _2\tau _1+\omega _2\tau _2\right) +\frac{2\beta _1\beta _2}{\omega _2}\mathrm{cos}\left( \omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&+\frac{\beta _1^2}{3\omega _2^2}\mathrm{sin}\left( 3\omega _2\tau _1\right) -\frac{\beta _1^2}{\omega _2^2}\mathrm{sin}\left( 2\omega _2\tau _1\right) -\frac{2\beta _2^2}{3}\mathrm{sin}\left( 3\omega _2\tau _2\right) \nonumber \\&+\frac{2\beta _2\gamma _5}{3\omega _2}\mathrm{cos}\left( 2\omega _2\tau _2\right) +\beta _2^2\mathrm{sin}\left( 2\omega _2\tau _2\right) -\beta _9\mathrm{sin}\left( \omega _2\tau _1\right) \nonumber \\&-\frac{2\beta _1\beta _2}{3\omega _2}\mathrm{cos}\left( \omega _2\tau _1+2\omega _2\tau _2\right) +\omega _2\beta _{15}\mathrm{cos}\left( \omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&+\omega _2\beta _{10}\mathrm{cos}\left( \omega _2\tau _2\right) +\omega _2^2\beta _{13}\mathrm{sin}\left( 2\omega _2\tau _2\right) -\beta _{14}\mathrm{sin}\left( 2\omega _2\tau _1\right) \nonumber \\ \end{aligned}$$

(C.11)

$$\begin{aligned} \varXi _6=&\frac{10\gamma _5^2}{3\omega _2^2}-\frac{5\gamma _5\beta _1}{\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) -\frac{5\gamma _5\beta _2}{\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) -3\gamma _7\nonumber \\&-\frac{2\beta _1\gamma _5}{\omega _2^2}+\frac{2\beta _1^2}{\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) +\frac{2\beta _1\beta _2}{\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) +\frac{2\beta _1^2}{3\omega _2^2}\nonumber \\&+\frac{\beta _1\gamma _5}{3\omega _2^2}\mathrm{cos}\left( 2\omega _2\tau _1\right) -\frac{\beta _1^2}{3\omega _2^2}\mathrm{cos}\left( 3\omega _2\tau _1\right) +\frac{\beta _1^2}{\omega _2^2}\mathrm{cos}\left( 2\omega _2\tau _1\right) \nonumber \\&-\frac{\beta _1\beta _2}{3\omega _2}\mathrm{sin}\left( 2\omega _2\tau _1+\omega _2\tau _2\right) +\frac{2\beta _1\beta _2}{\omega _2}\mathrm{sin}\left( \omega _2\tau _1+\omega _2\tau _2\right) \nonumber \\&+\frac{4\beta _1\beta _2}{3\omega _2}\mathrm{sin}\left( \omega _2\tau _2-\omega _2\tau _1\right) -\frac{2\beta _1\beta _2}{3\omega _2}\mathrm{sin}\left( \omega _2\tau _1+2\omega _2\tau _2\right) \nonumber \\&+\frac{2\beta _2\gamma _5}{3\omega _2}\mathrm{sin}\left( 2\omega _2\tau _2\right) +\frac{2\beta _2^2}{3}\mathrm{cos}\left( 3\omega _2\tau _2\right) -\beta _2^2\mathrm{cos}\left( 2\omega _2\tau _2\right) \nonumber \\&+\frac{2\beta _2^2}{3}+3\beta _9\mathrm{cos}\left( \omega _2\tau _1\right) +3\omega _2\beta _{10}\mathrm{sin}\left( \omega _2\tau _2\right) +2\omega _2^2\beta _{13}\nonumber \\&-\omega _2^2\beta {13}\mathrm{cos}\left( 2\omega _2\tau _2\right) +2\beta _{14}+2\omega _2\beta _{15}\mathrm{sin}\left( \omega _2\tau _2-\omega _2\tau _1\right) \nonumber \\&+\beta _{14}\mathrm{cos}\left( 2\omega _2\tau _1\right) +\omega _2\beta _{15}\mathrm{sin}\left( \omega _2\tau _1+\omega _2\tau _2\right) \end{aligned}$$

(C.12)

$$\begin{aligned} \varLambda _7=&\frac{2\gamma _4\alpha _1}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _1\right) -\beta _5\mathrm{sin}\left( \omega _2\tau _1\right) +\frac{\beta _1\gamma _4}{\omega _2^2}\mathrm{sin}\left( \omega _2\tau _1\right) \nonumber \\&-\frac{2\omega _2\gamma _4\alpha _2}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( \omega _2\tau _2\right) -\frac{\beta _2\gamma _4}{\omega _2}\mathrm{cos}\left( \omega _2\tau _2\right) \nonumber \\&+\omega _2\beta _6\mathrm{cos}\left( \omega _2\tau _2\right) \end{aligned}$$

(C.13)

$$\begin{aligned} \varXi _7=&\frac{2\gamma _1\gamma _4}{4\omega _1^2-\omega _2^2}-\frac{2\gamma _4\alpha _1}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) -\frac{4\gamma _4\alpha _2\omega _2}{2\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\frac{2\gamma _4\gamma _5}{\omega _2^2}-\frac{\beta _1\gamma _4}{\omega _2^2}-\frac{\beta _1\gamma _4}{\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) -\frac{\beta _2\gamma _4}{\omega _2}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\beta _{15}\mathrm{cos}\left( \omega _2\tau _1\right) +\omega _2\beta _6\mathrm{sin}\left( \omega _2\tau _2\right) +\omega _1\beta _7\mathrm{sin}\left( \omega _1\tau _2\right) \nonumber \\&+\beta _8\mathrm{cos}\left( \omega _1\tau _1\right) +\beta _{11}\omega _1^2+\omega _1\beta _{12}\mathrm{sin}\left( \omega _1\tau _2-\omega _1\tau _1\right) \nonumber \\&-\gamma _6+2\beta _{16} \end{aligned}$$

(C.14)

$$\begin{aligned} \varLambda _8=&\frac{2\gamma _4\alpha _1}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{sin}\left( \omega _2\tau _1\right) -\frac{2\omega _2\gamma _4\alpha _2}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{cos}\left( \omega _2\tau _2\right) \nonumber \\&-\frac{\beta _1\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( 2\omega _1\tau _1\right) +\frac{\beta _1\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _1\right) \nonumber \\&+\frac{2\omega _1\beta _2\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( 2\omega _1\tau _2\right) -\frac{\omega _2\beta _2\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( \omega _2\tau _2\right) \nonumber \\&+\beta _5\mathrm{sin}\left( \omega _2\tau _1\right) -\beta _6\omega _2\mathrm{cos}\left( \omega _2\tau _2\right) +\beta _7\omega _1\mathrm{cos}\left( \omega _1\tau _2\right) \nonumber \\&-\beta _8\mathrm{sin}\left( \omega _1\tau _1\right) +\beta _{11}\omega _1^2\mathrm{sin}\left( 2\omega _1\tau _2\right) -\beta _{16}\mathrm{sin}\left( 2\omega _1\tau _1\right) \nonumber \\&+\beta _{12}\omega _1\mathrm{cos}\left( \omega _1\tau _1+\omega _1\tau _2\right) \end{aligned}$$

(C.15)

$$\begin{aligned} \varXi _8=&\frac{2\gamma _4\alpha _1}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{cos}\left( \omega _2\tau _1\right) +\frac{2\omega _2\gamma _4\alpha _2}{\omega _2\left( \omega _2-2\omega _1\right) }\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\frac{\beta _1\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( 2\omega _1\tau _1\right) +\frac{\beta _1\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{cos}\left( \omega _2\tau _1\right) \nonumber \\&+\frac{2\omega _1\beta _2\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( 2\omega _1\tau _2\right) +\frac{\omega _2\beta _2\gamma _4}{4\omega _1^2-\omega _2^2}\mathrm{sin}\left( \omega _2\tau _2\right) \nonumber \\&+\frac{2\gamma _1\gamma _4}{\omega _2\left( 2\omega _1-\omega _2\right) }+\frac{2\gamma _4\gamma _5}{\omega _2^2-4\omega _1^2}-\gamma _6+\beta _5\mathrm{cos}\left( \omega _2\tau _1\right) \nonumber \\&+\beta _6\omega _2\mathrm{sin}\left( \omega _2\tau _2\right) +\beta _7\omega _1\mathrm{sin}\left( \omega _1\tau _2\right) +\beta _8\mathrm{sin}\left( \omega _1\tau _1\right) \nonumber \\&-\beta _{11}\omega _1^2\mathrm{cos}\left( 2\omega _1\tau _2\right) +\beta _{12}\omega _1\mathrm{sin}\left( \omega _1\tau _1+\omega _1\tau _2\right) \nonumber \\&+\beta _{16}\mathrm{cos}\left( 2\omega _1\tau _1\right) \end{aligned}$$

(C.16)

Appendix

$$\begin{aligned} R_{11}=&\varLambda _1\frac{1}{2\omega _1}+\varLambda _2\frac{3a_1^2}{8\omega _1}+\varLambda _3\frac{a_2^2}{4\omega _1}+\varLambda _4\frac{a_2^2}{8\omega _1}\nonumber \\&\times \mathrm{cos}\left( 2\psi _1-2\psi _2\right) +\varXi _4\frac{a_2^2}{8\omega _1}\mathrm{sin}\left( 2\psi _1-2\psi _2\right) \end{aligned}$$

(D.1)

$$\begin{aligned} R_{12}=&-\varLambda _4\frac{a_1a_2^2}{4\omega _1}\mathrm{sin}\left( 2\psi _1-2\psi _2\right) +\varXi _4\frac{a_1a_2^2}{4\omega _1}\nonumber \\&\times \mathrm{cos}\left( 2\psi _1-2\psi _2\right) +\frac{f\varOmega ^2}{\omega _1}\mathrm{cos}\psi _1 \end{aligned}$$

(D.2)

$$\begin{aligned} R_{13}=&\varLambda _4\frac{a_1a_2}{2\omega _1}\mathrm{cos}\left( 2\psi _1-2\psi _2\right) +\varXi _4\frac{a_1a_2}{4\omega _1}\nonumber \\&\times \mathrm{sin}\left( 2\psi _1-2\psi _2\right) +\varLambda _3\frac{a_1a_2}{2\omega _1} \end{aligned}$$

(D.3)

$$\begin{aligned} R_{14}=&\varLambda _4\frac{a_1a_2^2}{4\omega _1}\mathrm{sin}\left( 2\psi _1-2\psi _2\right) \nonumber \\&-\varXi _4\frac{a_1a_2^2}{4\omega _1}\mathrm{cos}\left( 2\psi _1-2\psi _2\right) \end{aligned}$$

(D.4)

$$\begin{aligned} R_{21}=&\varXi _2\frac{a_1}{4\omega _1}-\frac{f\varOmega ^2}{a_1^2\omega _1}\mathrm{cos}\psi _1 \end{aligned}$$

(D.5)

$$\begin{aligned} R_{22}=&-\varLambda _4\frac{a_2^2}{4\omega _1}\mathrm{cos}\left( 2\psi _1-2\psi _2\right) -\varXi _4\frac{a_2^2}{4\omega _1}\nonumber \\&\times \mathrm{sin}\left( 2\psi _1-2\psi _2\right) -\frac{f\varOmega ^2}{a_1\omega _1}\mathrm{sin}\psi _1 \end{aligned}$$

(D.6)

$$\begin{aligned} R_{23}=&-\varLambda _4\frac{a_2}{4\omega _1}\mathrm{sin}\left( 2\psi _1-2\psi _2\right) +\varXi _4\frac{a_2}{4\omega _1}\nonumber \\&\times \mathrm{cos}\left( 2\psi _1-2\psi _2\right) +\varXi _3\frac{a_2}{2\omega _1} \end{aligned}$$

(D.7)

$$\begin{aligned} R_{24}=&\varLambda _4\frac{a_2^2}{4\omega _1}\mathrm{cos}\left( 2\psi _1-2\psi _2\right) \nonumber \\&+\varXi _4\frac{a_2^2}{4\omega _1}\mathrm{sin}\left( 2\psi _1-2\psi _2\right) \end{aligned}$$

(D.8)

$$\begin{aligned} R_{31}=&\varLambda _8\frac{a_1a_2}{4\omega _2}\mathrm{cos}\left( 2\psi _2-2\psi _1\right) +\varXi _8\frac{a_1a_2}{4\omega _2}\nonumber \\&\times \mathrm{sin}\left( 2\psi _2-2\psi _1\right) +\varLambda _7\frac{a_1a_2}{2\omega _2} \end{aligned}$$

(D.9)

$$\begin{aligned} R_{32}=&\varLambda _8\frac{a_1^2a_2}{4\omega _2}\mathrm{sin}\left( 2\psi _2-2\psi _1\right) \nonumber \\&-\varXi _8\frac{a_1^2a_2}{4\omega _2}\left( 2\psi _2-2\psi _1\right) \end{aligned}$$

(D.10)

$$\begin{aligned} R_{33}=&\varLambda _5\frac{1}{2\omega _2}+\varLambda _6\frac{3a_2^2}{8\omega _2}+\varLambda _7\frac{a_1^2}{4\omega _2}+\varLambda _8\frac{a_1^2}{8\omega _2}\nonumber \\&\times \mathrm{cos}\left( 2\psi _2-2\psi _1\right) +\varXi _8\frac{a_1^2}{8\omega _2}\mathrm{sin}\left( 2\psi _2-2\psi _1\right) \end{aligned}$$

(D.11)

$$\begin{aligned} R_{34}=&-\varLambda _8\frac{a_1^2a_2}{4\omega _2}\mathrm{sin}\left( 2\psi _2-2\psi _1\right) +\varXi _8\frac{a_1^2a_2}{4\omega _2}\nonumber \\&\times \mathrm{cos}\left( 2\psi _2-2\psi _1\right) +\frac{f\varOmega ^2}{\omega _2}\mathrm{sin}\psi _2 \end{aligned}$$

(D.12)

$$\begin{aligned} R_{41}=&-\varLambda _8\frac{a_1}{4\omega _2}\mathrm{sin}\left( 2\psi _2-2\psi _1\right) +\varXi _8\frac{a_1}{4\omega _2}\nonumber \\&\times \mathrm{cos}\left( 2\psi _2-2\psi _1\right) +\varXi _7\frac{a_1}{2\omega _2} \end{aligned}$$

(D.13)

$$\begin{aligned} R_{42}=&\varLambda _8\frac{a_1^2}{4\omega _2}\mathrm{cos}\left( 2\psi _2-2\psi _1\right) +\varXi _8\frac{a_1^2}{4\omega _2}\mathrm{sin}\left( 2\psi _2-2\psi _1\right) \end{aligned}$$

(D.14)

$$\begin{aligned} R_{43}=&\varXi _6\frac{a_2}{4\omega _2}-\frac{f\varOmega ^2}{a_2^2\omega _2}\mathrm{sin}\psi _2 \end{aligned}$$

(D.15)

$$\begin{aligned} R_{44}=&-\varLambda _8\frac{a_1^2}{4\omega _2}\mathrm{cos}\left( 2\psi _2-2\psi _1\right) -\varXi _8\frac{a_1^2}{4\omega _2}\nonumber \\&\times \mathrm{sin}\left( 2\psi _2-2\psi _1\right) +\frac{f\varOmega ^2}{a_2\omega _2}\mathrm{cos}\psi _2 \end{aligned}$$

(D.16)