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Vibration suppression of a stay cable using a time-delayed nonlinear energy sink under multi-source external excitation

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Abstract

Stay cables can exhibit undesirable vibrations under external excitations. This paper studies the vibration control of a stay cable structure via a time-delayed nonlinear energy sink with nonlinear damping. The system dynamics are modeled separately for uniformly distributed load and support motion. The Galerkin method is used to discretize the original partial differential equations. The dynamic responses of the system under the two cases are obtained. The reasonable intervals for the control parameters are determined through a bifurcation analysis of the trivial solution. To elucidate the parametric influence, the effects of nonlinear damping, excitation amplitude, and frequency on the displacement amplitude are quantified through numerical analysis. The influences of the control parameters on the dynamic responses and energy of the system are explored. The genetic algorithm for parameter optimization is employed and the controller exhibits high performance with tunable parameters. The bifurcation diagram is given and the complicated dynamical features are observed. These results demonstrate that time-delayed feedback control plays a key role in improving the vibration reduction performance of nonlinear energy sink for cable system vibration mitigation, with specific manifestations including the suppression of resonant peaks and the alleviation of complicated dynamic behaviors in the stay cable. The novelty of this work lies in leveraging the interplay between these nonlinear elements to control the dynamics of the stay cable.

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Acknowledgements

The authors thank the anonymous reviewers for their helpful comments and suggestions that have helped to improve the presentation.

Funding

The authors declare that no funds were received during the preparation of this manuscript.

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Authors and Affiliations

  1. College of Mechanics and Engineering Science, Hohai University, 211100, Nanjing, China

    Zhiqun Liu & Weijie Ding

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  1. Zhiqun Liu
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  2. Weijie Ding
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Contributions

Zhiqun Liu: Data curation, Formal analysis, Investigation, Software, Validation, Writing - original draft. Weijie Ding: Conceptualization, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Supervision, Writing - review & editing.

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Correspondence to Weijie Ding.

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Appendices

Appendix A

The elements of \(\mathbf {\Psi }\) and \(\mathbf {\Theta }\) in Eq. (12) are

$$\begin{aligned} \mathbf {\chi }_{11}&={a_{15}} {d_{1}}^2 \varphi (l_1)/4 + {a_{15}} \varphi (l_1)^3 {b_{1}}^2/4 \nonumber \\&\quad + (3 {a_{15}} {c_{1}}^2 \varphi (l_1))/4 + (3 {a_{15}} \varphi (l_1)^3 {a_{1}}^2)/4 + \mu \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {b_{1}} {d_{1}}/2 - (3 {a_{15}} \varphi (l_1)^2 {a_{1}} {c_{1}})/2 \nonumber \\&\quad + {a_{18}} \varphi (l_1) \cos (\varpi {\gamma }) + {a_{16}} \varphi (l_1); \end{aligned}$$
(A.1)
$$\begin{aligned} \mathbf {\chi }_{12}&=2 \varpi + {a_{15}} \varphi (l_1)^3 {a_{1}} {b_{1}}/2 \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {b_{1}} {c_{1}}/2 - {a_{15}} \varphi (l_1)^2 {a_{1}} {d_{1}}/2 \nonumber \\&\quad + {a_{15}} {c_{1}} {d_{1}} \varphi (l_1)/2 \nonumber \\&\quad - {a_{18}} \varphi (l_1) \sin (\varpi {\gamma }); \end{aligned}$$
(A.2)
$$\begin{aligned} \mathbf {\chi }_{13}&=-{a_{15}} \varphi (l_1)^2 {b_{1}}^2/4 - (3 {a_{15}} \varphi (l_1)^2 {a_{1}}^2)/4 - {a_{18}} \cos (\varpi {\gamma }) \nonumber \\&\quad - {a_{16}} + {a_{15}} \varphi (l_1) {b_{1}} {d_{1}}/2 \nonumber \\&\quad + (3 {a_{15}} \varphi (l_1) {a_{1}} {c_{1}})/2 \nonumber \\&\quad - (3{a_{15}} {c_{1}}^2)/4 - {a_{15}} {d_{1}}^2/4; \end{aligned}$$
(A.3)
$$\begin{aligned} \mathbf {\chi }_{14}&=-{a_{15}} {c_{1}} {d_{1}}/2 + {a_{18}} \sin (\varpi {\gamma }) \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {a_{1}} {b_{1}}/2 + {a_{15}} \varphi (l_1) {b_{1}} {c_{1}}/2 \nonumber \\&\quad + {a_{15}} \varphi (l_1) {a_{1}} {d_{1}}/2; \end{aligned}$$
(A.4)
$$\begin{aligned} \mathbf {\chi }_{21}&=-{a_{15}} \varphi (l_1)^2 {a_{1}} {d_{1}}/2 - {a_{15}} \varphi (l_1)^2 {b_{1}} {c_{1}}/2 \nonumber \\&\quad + {a_{15}} {c_{1}} {d_{1}} \varphi (l_1)/2 \nonumber \\&\quad + {a_{15}} \varphi (l_1)^3 {a_{1}} {b_{1}}/2 - 2 \varpi \nonumber \\&\quad + {a_{18}} \varphi (l_1) \sin (\varpi {\gamma }); \end{aligned}$$
(A.5)
$$\begin{aligned} \mathbf {\chi }_{22}&=-(3 {a_{15}} \varphi (l_1)^2 {b_{1}} {d_{1}})/2 \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {a_{1}} {c_{1}}/2 + u + (3 {a_{15}} {d_{1}}^2 \varphi (l_1))/4 \nonumber \\&\quad + (3 {a_{15}} \varphi (l_1)^3 {b_{1}}^2)/4 \nonumber \\&\quad +(3 {a_{15}} \varphi (l_1)^3 {b_{1}}^2)/4 + {a_{15}} {c_{1}}^2 \varphi (l_1)/4 \nonumber \\&\quad + {a_{15}} \varphi (l_1)^3 {a_{1}}^2/4 + {a_{18}} \varphi (l_1) \cos (\varpi {\gamma }) + {a_{16}} \varphi (l_1); \end{aligned}$$
(A.6)
$$\begin{aligned} \mathbf {\chi }_{23}&={a_{15}} \varphi (l_1) {a_{1}} {d_{1}}/2 - {a_{15}} \varphi (l_1)^2 {a_{1}} {b_{1}}/2 \nonumber \\&\quad + {a_{15}} \varphi (l_1) {b_{1}} {c_{1}}/2 - {a_{18}} \sin (\varpi {\gamma }) \nonumber \\&\quad - {a_{15}} {c_{1}} {d_{1}}/2; \end{aligned}$$
(A.7)
$$\begin{aligned} \mathbf {\chi }_{24}&=(3 {a_{15}} \varphi (l_1) {b_{1}} {d_{1}})/2 + {a_{15}} \varphi (l_1) {a_{1}} {c_{1}}/2 \nonumber \\&\quad - {a_{16}} - {a_{18}} \cos (\varpi {\gamma }) \nonumber \\&\quad - (3 {a_{15}} \varphi (l_1)^2 {b_{1}}^2)/4 \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {a_{1}}^2/4 - {a_{15}} {c_{1}}^2/4 \nonumber \\&\quad - (3 {a_{15}} {d_{1}}^2)/4; \end{aligned}$$
(A.8)
$$\begin{aligned} \mathbf {\chi }_{31}&={\zeta _{n}} \varphi (l_1)^2 {b_{1}} {d_{1}}/2 + (3 {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {c_{1}})/2 \nonumber \\&\quad - {\zeta _{n}} {d_{1}}^2 \varphi (l_1)/4 - {\zeta _{n}} \varphi (l_1)^3 {b_{1}}^2/4 - (3 {\zeta _{n}} \nonumber \\&\quad \varphi (l_1)^3 {a_{1}}^2)/4- (3 {\zeta _{n}} {c_{1}}^2 \varphi (l_1))/4 \nonumber \\&\quad - {\zeta _{nl}} \varphi (l_1) - g_v \varphi (l_1) \cos (\varpi {\gamma }); \end{aligned}$$
(A.9)
$$\begin{aligned} \mathbf {\chi }_{32}&={\zeta _{n}} \varphi (l_1)^2 {a_{1}} {d_{1}}/2 - {\zeta _{n}} {c_{1}} {d_{1}} \varphi (l_1)/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {b_{1}} {c_{1}}/2 - {\zeta _{n}} \varphi (l_1)^3 {a_{1}} {b_{1}}/2 \nonumber \\&\quad + g_v \varphi (l_1) \sin (\varpi {\gamma }); \end{aligned}$$
(A.10)
$$\begin{aligned} \mathbf {\chi }_{33}&=-{\zeta _{n}} \varphi (l_1) {b_{1}} {d_{1}}/2 - (3 {\zeta _{n}} \varphi (l_1) {a_{1}} {c_{1}})/2 \nonumber \\&\quad + g_v \cos (\varpi {\gamma }) + {\zeta _{nl}} + (3 {\zeta _{n}} \varphi (l_1)^2 {a_{1}}^2)/4 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {b_{1}}^2/4 + (3 {\zeta _{n}} {c_{1}}^2)/4 + {\zeta _{n}} {d_{1}}^2/4; \end{aligned}$$
(A.11)
$$\begin{aligned} \mathbf {\chi }_{34}&=-{\zeta _{n}} \varphi (l_1) {a_{1}} {d_{1}}/2 - {\zeta _{n}} \varphi (l_1) {b_{1}} {c_{1}}/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {b_{1}}/2 + 2 \varpi - g_v \sin (\varpi {\gamma }) \nonumber \\&\quad + {\zeta _{n}} {c_{1}} {d_{1}}/2; \end{aligned}$$
(A.12)
$$\begin{aligned} \mathbf {\chi }_{41}&=-g_v \varphi (l_1) \sin (\varpi {\gamma }) - {\zeta _{n}} \varphi (l_1)^3 {a_{1}} {b_{1}}/2 + {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {d_{1}}/2 \nonumber \\&\quad - {\zeta _{n}} {c_{1}} {d_{1}} \varphi (l_1)/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {b_{1}} {c_{1}}/2; \end{aligned}$$
(A.13)
$$\begin{aligned} \mathbf {\chi }_{42}&=-{\zeta _{nl}} \varphi (l_1) - g_v \varphi (l_1) \cos (\varpi {\gamma }) + (3 {\zeta _{n}} \varphi (l_1)^2 {b_{1}} {d_{1}})/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {c_{1}}/2 \nonumber \\&\quad - (3 {\zeta _{n}} \varphi (l_1)^3 {b_{1}}^2)/4 - {\zeta _{n}} \varphi (l_1)^3 {a_{1}}^2/4 \nonumber \\&\quad - (3 {\zeta _{n}} {d_{1}}^2 \varphi (l_1))/4 \nonumber \\&\quad - {\zeta _{n}} {c_{1}}^2 \varphi (l_1)/4; \end{aligned}$$
(A.14)
$$\begin{aligned} \mathbf {\chi }_{43}&=-{\zeta _{n}} \varphi (l_1) {a_{1}} {d_{1}}/2 - {\zeta _{n}} \varphi (l_1) {b_{1}} {c_{1}}/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {b_{1}}/2 \nonumber \\&\quad + {\zeta _{n}} {c_{1}} {d_{1}}/2 - 2 \varpi + g_v \sin (\varpi {\gamma }); \end{aligned}$$
(A.15)
$$\begin{aligned} \mathbf {\chi }_{44}&={\zeta _{n}} {c_{1}}^2/4 + (3 {\zeta _{n}} {d_{1}}^2)/4 \nonumber \\&\quad - (3 {\zeta _{n}} \varphi (l_1) {b_{1}} {d_{1}})/2 - {\zeta _{n}} \varphi (l_1) {a_{1}} {c_{1}}/2 \nonumber \\&\quad + {\zeta _{nl}} + g_v \cos (\varpi {\gamma }) \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {a_{1}}^2/4 \nonumber \\&\quad + (3 {\zeta _{n}} \varphi (l_1)^2 {b_{1}}^2)/4; \end{aligned}$$
(A.16)
$$\begin{aligned} \mathbf {\Theta }_{1}&={a_{11}} {a_{1}} - {a_{1}} \varpi ^2 - {a_{17}} \nonumber \\&\quad + {a_{16}} \varphi (l_1) {b_{1}} \varpi \nonumber \\&\quad - {a_{18}} {c_{1}} \varpi \sin (\varpi {\gamma })\nonumber \\&\quad - {a_{18}} {d_{1}} \varpi \cos (\varpi {\gamma }) + \mu {b_{1}} \varpi - {a_{16}} {d_{1}} \varpi \nonumber \\&\quad + {a_{18}} \varphi (l_1) {a_{1}} \varpi \sin (\varpi {\gamma }) + {a_{18}} \varphi (l_1) {b_{1}} \varpi \cos (\varpi {\gamma }) \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1) {b_{1}} {c_{1}} {d_{1}})/2 \nonumber \\&\quad - (3 {a_{15}} \varphi (l_1)^2 {b_{1}}^2 {d_{1}} \varpi )/4 - (3 {a_{14}} \varphi (l_1)^2 {a_{1}} {b_{1}} {d_{1}})/2 \nonumber \\&\quad + (3 {a_{15}} \varphi (l_1) {b_{1}} {d_{1}}^2 \varpi )/4 \nonumber \\&\quad + {a_{15}} {c_{1}}^2 \varphi (l_1) {b_{1}} \varpi /4 {a_{15}} \varphi (l_1)^2 {a_{1}}^2 {d_{1}} \varpi /4 \nonumber \\&\quad + {a_{15}} \varphi (l_1)^3 {a_{1}}^2 {b_{1}} \varpi /4 - (3 {a_{14}} {c_{1}} {d_{1}}^2)/4 \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1)^3 {a_{1}}^3)/4 - {a_{15}} {d_{1}}^3 \varpi /4 \nonumber \\&\quad + (3 {a_{13}} {a_{1}} {b_{1}}^2)/4 + (3 {a_{14}} \varphi (l_1) {a_{1}} {d_{1}}^2)/4 \nonumber \\&\quad - {a_{15}} {c_{1}}^2 {d_{1}} \varpi /4 - (3 {a_{14}} \varphi (l_1)^2 {b_{1}}^2 {c_{1}})/4 \nonumber \\&\quad - (9 {a_{14}} \varphi (l_1)^2 {a_{1}}^2 {c_{1}})/4+(3 {a_{14}} \varphi (l_1)^2 {b_{1}}^2 {c_{1}})/4 \nonumber \\&\quad - (9 {a_{14}} \varphi (l_1)^2 {a_{1}}^2 {c_{1}})/4 + (9 {a_{14}} \varphi (l_1) {a_{1}} {c_{1}}^2)/4 \nonumber \\&\quad + {a_{15}} \varphi (l_1)^3 {b_{1}}^3 \varpi /4 \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1)^3 {a_{1}} {b_{1}}^2)/4 - (3 {a_{14}} {c_{1}}^3)/4 \nonumber \\&\quad - {a_{15}} \varphi (l_1)^2 {a_{1}} {c_{1}} {b_{1}} \varpi /2 \nonumber \\&\quad + {a_{15}} \varphi (l_1) {a_{1}} {c_{1}} {d_{1}} \varpi /2 + (3 {a_{13}} {a_{1}}^3)/4; \end{aligned}$$
(A.17)
$$\begin{aligned} \mathbf {\Theta }_{2}&={a_{15}} {c_{1}}^3 \varpi /4 + (3 {a_{13}} {a_{1}}^2 {b_{1}})/4 \nonumber \\&\quad - (3 {a_{14}} {c_{1}}^2 {d_{1}})/4 - {b_{1}} \varpi ^2 + {a_{11}} {b_{1}} + {a_{15}} {c_{1}} {d_{1}}^2 \varpi /4 \nonumber \\&\quad + (3 {a_{13}} {b_{1}}^3)/4- (3 {a_{14}} {d_{1}}^3)/4 \nonumber \\&\quad + {a_{15}} \varphi (l_1)^2 {b_{1}} {d_{1}} {a_{1}} \varpi /2 \nonumber \\&\quad - {a_{15}} \varphi (l_1) {b_{1}} {d_{1}} {c_{1}} \varpi /2 \nonumber \\&\quad - {a_{15}} \varphi (l_1)^3 {a_{1}}^3 \varpi /4 + (3 {a_{14}} \varphi (l_1)^3 {a_{1}}^2 {b_{1}})/4 \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1) {b_{1}} {c_{1}}^2)/4 - (3 {a_{14}} \varphi (l_1)^2 {a_{1}}^2 {d_{1}})/4 \nonumber \\&\quad - (3 {a_{14}} \varphi (l_1)^2 {a_{1}}^2 {d_{1}})/4 \nonumber \\&\quad - (9 {a_{14}} \varphi (l_1)^2 {b_{1}}^2 {d_{1}})/4 + (9 {a_{14}} \varphi (l_1) {b_{1}} {d_{1}}^2)/4 \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1)^3 {b_{1}}^3)/4 - {a_{15}} \varphi (l_1)^3 {b_{1}}^2 {a_{1}} \varpi /4 \nonumber \\&\quad - (3 {a_{15}} \varphi (l_1) {a_{1}} {c_{1}}^2 \varpi )/4 - {a_{16}} \varphi (l_1) {a_{1}} \varpi \nonumber \\&\quad + {a_{18}} {c_{1}} \varpi \cos (\varpi {\gamma }) - {a_{18}} {d_{1}} \varpi \sin (\varpi {\gamma }) \nonumber \\&\quad - \mu {a_{1}} \varpi + {a_{16}} {c_{1}} \varpi + {a_{15}} \varphi (l_1)^2 {b_{1}}^2 {c_{1}} \varpi /4\nonumber \\&\quad - (3 {a_{14}} \varphi (l_1)^2 {a_{1}} {b_{1}} {c_{1}})/2 + (3 {a_{15}} \varphi (l_1)^2 {a_{1}}^2 {c_{1}} \varpi )/4 \nonumber \\&\quad - {a_{18}} \varphi (l_1) {a_{1}} \varpi \cos (\varpi {\gamma }) \nonumber \\&\quad + {a_{18}} \varphi (l_1) {b_{1}} \varpi \sin (\varpi {\gamma }) - {a_{15}} {d_{1}}^2 \varphi (l_1) {a_{1}} \varpi /4 \nonumber \\&\quad + (3 {a_{14}} \varphi (l_1) {a_{1}} {c_{1}} {d_{1}})/2; \end{aligned}$$
(A.18)
$$\begin{aligned} \mathbf {\Theta }_{3}&= (3 {k_n} {c_{1}}^3)/4 - {\zeta _{n}} {c_{1}}^2 \varphi (l_1) \varpi {b_{1}}/4 - \varpi ^2 {c_{1}} \nonumber \\&\quad - {\zeta _{n}} \varphi (l_1)^3 {a_{1}}^2 \varpi {b_{1}}/4 + {\zeta _{n}} {d_{1}}^3 \varpi /4 \nonumber \\&\quad - (3 {k_n} \varphi (l_1)^3 {a_{1}}^3)/4+ (3 {k_n} {c_{1}} {d_{1}}^2)/4 + {\zeta _{n}} {c_{1}}^2 {d_{1}} \varpi /4 \nonumber \\&\quad - (3 {k_n} \varphi (l_1) {a_{1}} {d_{1}}^2)/4 \nonumber \\&\quad + (3 {k_n} \varphi (l_1)^2 {b_{1}}^2 {c_{1}})/4 \nonumber \\&\quad - (3 {\zeta _{n}} \varphi (l_1) {b_{1}} {d_{1}}^2 \varpi )/4 + (3 {\zeta _{n}} \varphi (l_1) {b_{1}} {d_{1}}^2 \varpi )/4 \nonumber \\&\quad + (3 {\zeta _{n}} \varphi (l_1)^2 {b_{1}}^2 {d_{1}} \varpi )/4 + {\zeta _{n}} \varphi (l_1)^2 {a_{1}}^2 {d_{1}} \varpi /4 \nonumber \\&\quad + (9 {k_n} \varphi (l_1)^2 {a_{1}}^2 {c_{1}})/4 \nonumber \\&\quad - (9 {k_n} \varphi (l_1) {a_{1}} {c_{1}}^2)/4 - (3 {k_n} \varphi (l_1)^3 {a_{1}} {b_{1}}^2)/4 \nonumber \\&\quad + (3 {k_n} \varphi (l_1)^2 {a_{1}} {b_{1}} {d_{1}})/2 \nonumber \\&\quad + {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {b_{1}} {c_{1}} \varpi /2 - {\zeta _{n}} \varphi (l_1) {a_{1}} {d_{1}} {c_{1}} \varpi /2 \nonumber \\&\quad - {\zeta _{n}} \varphi (l_1)^3 {b_{1}}^3 \varpi /4 - {\zeta _{nl}} \varphi (l_1) \varpi {b_{1}} \nonumber \\&\quad + g_v {c_{1}} \varpi \sin (\varpi {\gamma })+ g_v {d_{1}} \varpi \cos (\varpi {\gamma }) + {\zeta _{nl}} {d_{1}} \varpi \nonumber \\&\quad - g_v \varphi (l_1) {a_{1}} \varpi \sin (\varpi {\gamma }) \nonumber \\&\quad - g_v \varphi (l_1) {b_{1}} \varpi \cos (\varpi {\gamma }) \nonumber \\&\quad - (3 {k_n} \varphi (l_1) {b_{1}} {c_{1}} {d_{1}})/2; \end{aligned}$$
(A.19)
$$\begin{aligned} \mathbf {\Theta }_{4}&= -(3 {k_n} \varphi (l_1) {b_{1}} {c_{1}}^2)/4 \nonumber \\&\quad + (3 {k_n} \varphi (l_1)^2 {a_{1}}^2 {d_{1}})/4 - {d_{1}} \varpi ^2 \nonumber \\&\quad + g_v \varphi (l_1) {a_{1}} \varpi \cos (\varpi {\gamma }) \nonumber \\&\quad - g_v \varphi (l_1) {b_{1}} \varpi \sin (\varpi {\gamma }) - (9 {k_n} \varphi (l_1) {b_{1}} {d_{1}}^2)/4 \nonumber \\&\quad + (9 {k_n} \varphi (l_1)^2 {b_{1}}^2 {d_{1}})/4 - {\zeta _{n}} {c_{1}} {d_{1}}^2 \varpi /4 \nonumber \\&\quad - {\zeta _{n}} {c_{1}}^3 \varpi /4 + (3 {k_n} {c_{1}}^2 {d_{1}})/4 \nonumber \\&\quad - (3 {k_n} \varphi (l_1)^3 {b_{1}}^3)/4 + {\zeta _{nl}} {a_{1}} \varphi (l_1) \varpi \nonumber \\&\quad - {\zeta _{nl}} {c_{1}} \varpi \nonumber \\&\quad - g_v {c_{1}} \varpi \cos (\varpi {\gamma }) + g_v {d_{1}} \varpi \sin (\varpi {\gamma })\nonumber \\&\quad + (3 {k_n} {d_{1}}^3)/4 + {\zeta _{n}} \varphi (l_1) {b_{1}} {c_{1}} {d_{1}} \varpi /2 \nonumber \\&\quad - {\zeta _{n}} \varphi (l_1)^2 {a_{1}} {b_{1}} {d_{1}} \varpi /2 \nonumber \\&\quad + (3 {k_n} \varphi (l_1)^2 {a_{1}} {b_{1}} {c_{1}})/2 \nonumber \\&\quad + {\zeta _{n}} {d_{1}}^2 {a_{1}} \varphi (l_1) \varpi /4\nonumber \\&\quad - (3 {k_n} \varphi (l_1) {a_{1}} {c_{1}} {d_{1}})/2 + {\zeta _{n}} \varphi (l_1)^3 {a_{1}}^3 \varpi /4 \nonumber \\&\quad - (3 {k_n} \varphi (l_1)^3 {a_{1}}^2 {b_{1}})/4\nonumber \\&\quad + (3 {k_n} \varphi (l_1) {a_{1}} {c_{1}} {d_{1}})/2 + {\zeta _{n}} \varphi (l_1)^3 {a_{1}}^3 \varpi /4\nonumber \\&\quad - (3 {k_n} \varphi (l_1)^3 {a_{1}}^2 {b_{1}})/4 + {\zeta _{n}} \varphi (l_1)^3 {b_{1}}^2 {a_{1}} \varpi /4 \nonumber \\&\quad + (3 {\zeta _{n}} \varphi (l_1) {a_{1}} {c_{1}}^2 \varpi )/4 \nonumber \\&\quad - (3 {\zeta _{n}} \varphi (l_1)^2 {a_{1}}^2 {c_{1}} \varpi )/4 \nonumber \\&\quad - {\zeta _{n}} \varphi (l_1)^2 {b_{1}}^2 {c_{1}} \varpi /4; \end{aligned}$$
(A.20)

Appendix B

The elements of \(\mathbf {\Xi }\) and \(\mathbf {\Sigma }\) in Eq. (13) are

$$\begin{aligned} \mathbf {\psi }_{11}&=\mu + a_{16} \varphi (l_1) + a_{18} \varphi (l_1) \cos (\varpi \gamma ) \nonumber \\&- 2 a_{15} \varphi (l_1)^2 r_{1} r_2 + (3 a_{15} \varphi (l_1) c_{1}^2)/4 + a_{15} \varphi (l_1)^3 b_{1}^2/4 \nonumber \\&+ (3 a_{15} \varphi (l_1)^3 a_{1}^2)/4 + a_{15} \varphi (l_1) d_{1}^2/4 \nonumber \\&+ a_{15} \varphi (l_1)^3 r_{1}^2 + a_{15} \varphi (l_1) r_{2}^2 + a_{15} \varphi (l_1) f_{sp}^2 l1^2 \cos (2 \alpha )/8 \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1} d_{1}/2 + a_{15} \varphi (l_1) f_{sp}^2 l_1^2/8 \nonumber \\&- (3 a_{15} \varphi (l_1)^2 a_{1} c_{1})/2 \nonumber \\&- a_{15} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha )/2 \nonumber \\&+ a_{15} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.1)
$$\begin{aligned} \mathbf {\psi }_{12}&=2 \varpi - a_{18} \varphi (l_1) \sin (\varpi \gamma ) \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} d_{1}/2 + a_{15} \varphi (l_1) c_{1} d_{1}/2 \nonumber \\&+ a_{15} \varphi (l_1)^3 a_{1} b_{1}/2 \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1} c_{1}/2 - a_{15} \varphi (l_1) f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&+ a_{15} \varphi (l_1)^2 a_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.2)
$$\begin{aligned} \mathbf {\psi }_{13}&=-a_{16} - a_{18} \cos (\varpi \gamma ) - (3 a_{15} c_{1}^2)/4 - a_{15} d_{1}^2/4 \nonumber \\&- a_{15} r_{2}^2 + 2 a_{15} \varphi (l_1) r_{1} r_{2} - a_{15} \varphi (l_1)^2 b_{1}^2/4 \nonumber \\&- (3 a_{15} \varphi (l_1)^2 a_{1}^2)/4 - a_{15} f_{sp}^2 l_1^2/8 - a_{15} f_{sp}^2 l_1^2 \cos (2 \alpha )/8 \nonumber \\&- a_{15} \varphi (l_1)^2 r_{1}^2 + a_{15} f_{sp} l_1 d_{1} \cos (\alpha )/2\nonumber \\&+ a_{15} \varphi (l_1) b_{1} d_{1}/2 + (3 a_{15} \varphi (l_1) a_{1} c_{1})/2 \nonumber \\&- a_{15} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.3)
$$\begin{aligned} \mathbf {\psi }_{14}&=a_{18} \sin (\varpi \gamma ) \nonumber \\&+ a_{15} \varphi (l_1) a_{1} d_{1}/2 - a_{15} c_{1} d_{1}/2 + a_{15} f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} b_{1}/2 \nonumber \\&+ a_{15} \varphi (l_1) b_{1} c_{1}/2 \nonumber \\&- a_{15} \varphi (l_1) a_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.4)
$$\begin{aligned} \mathbf {\psi }_{15}&=2 a_{1} a_{15} \varphi (l_1)^3 r_{1} \nonumber \\&- 2 a_{1} a_{15} \varphi (l_1)^2 r_{2} \nonumber \\&- 2 a_{15} c_{1} \varphi (l_1)^2 r_{1}\nonumber \\&+ 2 a_{15} c_{1} \varphi (l_1) r_{2}; \end{aligned}$$
(B.5)
$$\begin{aligned} \mathbf {\psi }_{16}&=-2 a_{1} a_{15} \varphi (l_1)^2 r_{1} \nonumber \\&+ 2 a_{1} a_{15} \varphi (l_1) r_{2} \nonumber \\&+ 2 a_{15} c_{1} \varphi (l_1) r_{1} \nonumber \\&- 2 a_{15} c_{1} r_{2}; \end{aligned}$$
(B.6)
$$\begin{aligned} \mathbf {\psi }_{21}&=-2 \varpi + a_{18} \varphi (l_1) \sin (\varpi \gamma ) \nonumber \\&+ a_{15} \varphi (l_1) c_{1} d_{1}/2 + a_{15} \varphi (l_1)^3 a_{1} b_{1}/2 \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1} c_{1}/2\nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} d_{1}/2 - a_{15} \varphi (l_1) f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&+ a_{15} \varphi (l_1)^2 a_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.7)
$$\begin{aligned} \mathbf {\psi }_{22}&=\mu + a_{16} \varphi (l_1) + a_{18} \varphi (l_1) \cos (\varpi \gamma ) \nonumber \\&+ a_{15} \varphi (l_1) c_{1}^2/4 + (3 a_{15} \varphi (l_1)^3 b_{1}^2)/4 \nonumber \\&+ a_{15} \varphi (l_1)^3 a_{1}^2/4\nonumber \\&+ (3 a_{15} \varphi (l_1) d_{1}^2)/4 + a_{15} \varphi (l_1)^3 r_{1}^2 \nonumber \\&+ a_{15} \varphi (l_1) r_{2}^2 + (3 a_{15} \varphi (l_1) f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- (3 a_{15} \varphi (l_1)^2 b_{1} d_{1})/2 \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} c_{1}/2 \nonumber \\&- 2 a_{15} \varphi (l_1)^2 r_{1} r_{2}\nonumber \\&+ (3 a_{15} \varphi (l_1) f_{sp}^2 l_1^2)/8 \nonumber \\&- (3 a_{15} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ (3 a_{15} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha ))/2; \end{aligned}$$
(B.8)
$$\begin{aligned} \mathbf {\psi }_{23}=&-a_{18} \sin (\varpi \gamma ) \nonumber \\&- a_{15} c_{1} d_{1}/2 + a_{15} f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} b_{1}/2 \nonumber \\&+ a_{15} \varphi (l_1) b_{1} c_{1}/2 + a_{15} \varphi (l_1) a_{1} d_{1}/2 \nonumber \\&- a_{15} \varphi (l_1) a_{1} f_{sp} l_1 \cos (\alpha )/2; \end{aligned}$$
(B.9)
$$\begin{aligned}&\mathbf {\psi }_{24}=-a_{16} - a_{18} \cos (\varpi \gamma ) \nonumber \\&- a_{15} r_{2}^2 - (3 a_{15} d_{1}^2)/4 - a_{15} c_{1}^2/4 \nonumber \\&- (3 a_{15} \varphi (l_1)^2 b_{1}^2)/4 - (3 a_{15} f_{sp}^2 l_1^2)/8\nonumber \\&- a_{15} \varphi (l_1)^2 a_{1}^2/4 - (3 a_{15} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- a_{15} \varphi (l_1)^2 r_{1}^2 + (3 a_{15} f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ (3 a_{15} \varphi (l_1) b_{1} d_{1})/2 + a_{15} \varphi (l_1) a_{1} c_{1}/2 \nonumber \\&+ 2 a_{15} \varphi (l_1) r_{1} r_{2} - (3 a_{15} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha ))/2; \end{aligned}$$
(B.10)
$$\begin{aligned}&\mathbf {\psi }_{25}=2 a_{15} \varphi (l_1)^3 r_{1} b_{1} - 2 a_{15} \varphi (l_1)^2 r_{1} d_{1} \nonumber \\&- 2 a_{15} \varphi (l_1)^2 b_{1} r_{2} + 2 a_{15} \varphi (l_1) r_{2} d_{1} \nonumber \\&- 2 a_{15} \varphi (l_1) f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&+ 2 a_{15} \varphi (l_1)^2 r_{1} f_{sp} l_1 \cos (\alpha ); \end{aligned}$$
(B.11)
$$\begin{aligned}&\mathbf {\psi }_{26}=-2 a_{15} r_{2} d_{1} + 2 a_{15} f_{sp} l_1 r_{2} \cos (\alpha )\nonumber \\&- 2 a_{15} \varphi (l_1)^2 r_{1} b_{1} + 2 a_{15} \varphi (l_1) r_{1} d_{1} + 2 a_{15} \varphi (l_1) b_{1} r_{2} \nonumber \\&- 2 a_{15} \varphi (l_1) r_{1} f_{sp} l_1 \cos (\alpha ); \end{aligned}$$
(B.12)
$$\begin{aligned}&\mathbf {\psi }_{31}=-\xi _{n} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha )/2 \nonumber \\&+ \xi _{n} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha )/2 - \xi _{n} \varphi (l_1) f_{sp}^2 l_1^2/8 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 b_{1} d_{1}/2 \nonumber \\&+ 2 \xi _{n} \varphi (l_1)^2 r_{1} r_{2} - g_v \varphi (l_1) \cos (\varpi \gamma ) \nonumber \\&- \xi _{nl} \varphi (l_1) - (3 \xi _{n} \varphi (l_1)^3 a_{1}^2)/4 - (3 \xi _{n} \varphi (l_1) c_{1}^2)/4 \nonumber \\&- \xi _{n} \varphi (l_1) d_{1}^2/4 - \xi _{n} \varphi (l_1)^3 r_{1}^2 \nonumber \\&- \xi _{n} \varphi (l_1) r_{2}^2 - \xi _{n} \varphi (l_1)^3 b_{1}^2/4\nonumber \\&- \xi _{n} \varphi (l_1) f_{sp}^2 l_1^2 \cos (2 \alpha )/8\nonumber \\&+ (3 \xi _{n} \varphi (l_1)^2 a_{1} c_{1})/2; \end{aligned}$$
(B.13)
$$\begin{aligned}&\mathbf {\psi }_{32}=\xi _{n} \varphi (l_1) f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&- \xi _{n} \varphi (l_1)^2 a_{1} f_{sp} l_1 \cos (\alpha )/2 - \xi _{n} \varphi (l_1)^3 a_{1} b_{1}/2 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1} d_{1}/2 \nonumber \\&+ g_v \varphi (l_1) \sin (\varpi \gamma ) - \xi _{n} \varphi (l_1) c_{1} d_{1}/2 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 b_{1} c_{1}/2; \end{aligned}$$
(B.14)
$$\begin{aligned}&\mathbf {\psi }_{33}=\xi _{nl} + g_v \cos (\varpi \gamma ) \nonumber \\&+ \xi _{n} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha )/2 - \xi _{n} \varphi (l_1) b_{1} d_{1}/2 \nonumber \\&- 2 \xi _{n} \varphi (l_1) r_{1} r_{2} \nonumber \\ &- \xi _{n} f_{sp} l_1 d_{1} \cos (\alpha )/2 \nonumber \\&+ (3 \xi _{n} c_{1}^2)/4 + \xi _{n} r_{2}^2 + \xi _{n} d_{1}^2/4 \nonumber \\&+ \xi _{n} f_{sp}^2 l_1^2/8 + (3 \xi _{n} \varphi (l_1)^2 a_{1}^2)/4 \nonumber \\&+ \xi _{n} f_{sp}^2 l_1^2 \cos (2 \alpha )/8 + \xi _{n} \varphi (l_1)^2 r_{1}^2 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 b_{1}^2/4 - (3 \xi _{n} \varphi (l_1) a_{1} c_{1})/2; \end{aligned}$$
(B.15)
$$\begin{aligned}&\mathbf {\psi }_{34}=2 \varpi - g_v \sin (\varpi \gamma ) \nonumber \\&+ \xi _{n} \varphi (l_1) a_{1} f_{sp} l_1 \cos (\alpha )/2 - \xi _{n} f_{sp} l_1 c_{1} \cos (\alpha )/2\nonumber \\ &+ \xi _{n} \varphi (l_1)^2 a_{1} b_{1}/2 \nonumber \\&- \xi _{n} \varphi (l_1) a_{1} d_{1}/2 + \xi _{n} c_{1} d_{1}/2 \nonumber \\&- \xi _{n} \varphi (l_1) b_{1} c_{1}/2; \end{aligned}$$
(B.16)
$$\begin{aligned}&\mathbf {\psi }_{35}=-2 a_{1} \varphi (l_1)^3 r_{1} \xi _{n} + 2 a_{1} \varphi (l_1)^2 r_{2} \xi _{n} \nonumber \\&+ 2 c_{1} \varphi (l_1)^2 r_{1} \xi _{n}\nonumber \\&- 2 c_{1} \varphi (l_1) r_{2} \xi _{n}; \end{aligned}$$
(B.17)
$$\begin{aligned}&\mathbf {\psi }_{36}=2 a_{1} \varphi (l_1)^2 r_{1} \xi _{n} - 2 a_{1} \varphi (l_1) r_{2} \xi _{n} \nonumber \\&- 2 c_{1} \varphi (l_1) r_{1} \xi _{n} + 2 c_{1} r_{2} \xi _{n}; \end{aligned}$$
(B.18)
$$\begin{aligned}&\mathbf {\psi }_{41}=\xi _{n} \varphi (l_1) f_{sp} l_1 c_{1} \cos (\alpha )/2 \nonumber \\&- \xi _{n} \varphi (l_1)^2 a_{1} f_{sp} l_1 \cos (\alpha )/2 - g_v \varphi (l_1) \sin (\varpi \gamma ) \nonumber \\&+\xi _{n} \varphi (l_1)^2 b_{1} c_{1}/2 - \xi _{n} \varphi (l_1)^3 a_{1} b_{1}/2 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1} d_{1}/2 - \xi _{n} \varphi (l_1) c_{1} d_{1}/2; \end{aligned}$$
(B.19)
$$\begin{aligned}&\mathbf {\psi }_{42}=-(3 \xi _{n} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha ))/2 \nonumber \\&+ (3 \xi _{n} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha ))/2 - g_v \varphi (l_1) \cos (\varpi \gamma ) \nonumber \\&- \xi _{nl} \varphi (l_1) - \xi _{n} \varphi (l_1)^3 r_{1}^2 - \xi _{n} \varphi (l_1) r_{2}^2 - (3 \xi _{n} \varphi (l_1)^3 b_{1}^2)/4 \nonumber \\&- \xi _{n} \varphi (l_1)^3 a_{1}^2/4 - \xi _{n} \varphi (l_1) c_{1}^2/4\nonumber \\&- (3 \xi _{n} \varphi (l_1) d_{1}^2)/4 + 2 \xi _{n} \varphi (l_1)^2 r_{1} r_{2} \nonumber \\&- (3 \xi _{n} \varphi (l_1) f_{sp}^2 l_1^2)/8 + (3 \xi _{n} \varphi (l_1)^2 b_{1} d_{1})/2 \nonumber \\&- (3 \xi _{n} \varphi (l_1) f_{sp}^2 l_1^2 \cos (2\alpha ))/8 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1} c_{1}/2; \end{aligned}$$
(B.20)
$$\begin{aligned}&\mathbf {\psi }_{43}=-2 \varpi + g_v \sin (\varpi \gamma ) + \xi _{n} \varphi (l_1) a_{1} f_{sp} l_1 \cos (\alpha )/2 \nonumber \\&+ \xi _{n} c_{1} d_{1}/2 - \xi _{n} \varphi (l_1) b_{1} c_{1}/2\nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1} b_{1}/2 \nonumber \\&- \xi _{n} \varphi (l_1) a_{1} d_{1}/2\nonumber \\&- \xi _{n} f_{sp} l_1 c_{1} \cos (\alpha )/2; \end{aligned}$$
(B.21)
$$\begin{aligned}&\mathbf {\psi }_{44}=(3 \xi _{n} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha ))/2 \nonumber \\&+ \xi _{nl} + g_v \cos (\varpi \gamma ) + \xi _{n} r_{2}^2 + \xi _{n} c_{1}^2/4 + (3 \xi _{n} d_{1}^2)/4 \nonumber \\&+ (3 \xi _{n} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 + \xi _{n} \varphi (l_1)^2 r_{1}^2 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1}^2/4 + (3 \xi _{n} \varphi (l_1)^2 b_{1}^2)/4 \nonumber \\&+ (3 \xi _{n} f_{sp}^2 l_1^2)/8 \nonumber \\&- 2 \xi _{n} \varphi (l_1) r_{1} r_{2} - (3 \xi _{n} \varphi (l_1) b_{1} d_{1})/2 - \xi _{n} \varphi (l_1) a_{1} c_{1}/2 \nonumber \\&- (3 \xi _{n} f_{sp} l_1 d_{1} \cos (\alpha ))/2; \end{aligned}$$
(B.22)
$$\begin{aligned}&\mathbf {\psi }_{45}=2 \xi _{n} \varphi (l_1) f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&- 2 \xi _{n} \varphi (l_1)^2 r_{1} f_{sp} l_1 \cos (\alpha ) - 2 \xi _{n} \varphi (l_1) r_{2} d_{1} \nonumber \\&+ 2 \xi _{n} \varphi (l_1)^2 b_{1} r_{2} \nonumber \\&+ 2 \xi _{n} \varphi (l_1)^2 r_{1} d_{1} - 2 \xi _{n} \varphi (l_1)^3 r_{1} b_{1}; \end{aligned}$$
(B.23)
$$\begin{aligned}&\mathbf {\psi }_{46}=2 \xi _{n} \varphi (l_1) r_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&+ 2 \xi _{n} r_{2} d_{1} + 2 \xi _{n} \varphi (l_1)^2 r_{1} b_{1} - 2 \xi _{n} \varphi (l_1) b_{1} r_{2} \nonumber \\&- 2 \xi _{n} f_{sp} l_1 r_{2} \cos (\alpha )\nonumber \\&- 2 \xi _{n} \varphi (l_1) r_{1} d_{1}; \end{aligned}$$
(B.24)
$$\begin{aligned}&\mathbf {\psi }_{51}=a_{1} a_{15} \varphi (l_1)^3 r_{1} - a_{1} a_{15} \varphi (l_1)^2 r_{2} \nonumber \\&- a_{15} c_{1} \varphi (l_1)^2 r_{1} \nonumber \\&+ a_{15} c_{1} \varphi (l_1) r_{2}; \end{aligned}$$
(B.25)
$$\begin{aligned}&\mathbf {\psi }_{52}=-a_{15} \varphi (l_1) f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&+ a_{15} \varphi (l_1)^2 r_{1} f_{sp} l_1 \cos (\alpha )\nonumber \\&+ a_{15} \varphi (l_1) r_{2} d_{1} + a_{15} \varphi (l_1)^3 r_{1} b_{1} \nonumber \\&- a_{15} \varphi (l_1)^2 r_{1} d_{1} \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1} r_{2}; \end{aligned}$$
(B.26)
$$\begin{aligned}&\mathbf {\psi }_{53}=-a_{1} a_{15} \varphi (l_1)^2 r_{1} \nonumber \\&+ a_{1} a_{15} \varphi (l_1) r_{2} + a_{15} c_{1} \varphi (l_1) r_{1} \nonumber \\&- a_{15} c_{1} r_{2}; \end{aligned}$$
(B.27)
$$\begin{aligned}&\mathbf {\psi }_{54}=-a_{15} r_{2} d_{1} - a_{15} \varphi (l_1) r_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&- a_{15} \varphi (l_1)^2 r_{1} b_{1} + a_{15} \varphi (l_1) r_{1} d_{1} \nonumber \\&+ a_{15} \varphi (l_1) b_{1} r_{2} \nonumber \\&+ a_{15} f_{sp} l_1 r_{2} \cos (\alpha ); \end{aligned}$$
(B.28)
$$\begin{aligned}&\mathbf {\psi }_{55}=a_{18} \varphi (l_1) + a_{16} \varphi (l_1) + \mu \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} c_{1} + a_{15} \varphi (l_1) f_{sp}^2 l_1^2/4 \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1} d_{1} \nonumber \\&+ a_{15} \varphi (l_1) d_{1}^2/2 + a_{15} \varphi (l_1) c_{1}^2/2 + a_{15} \varphi (l_1)^3 a_{1}^2/2 \nonumber \\&+ a_{15} \varphi (l_1)^3 b_{1}^2/2 + a_{15} \varphi (l_1)^3 r_{1}^2 \nonumber \\&+ a_{15} \varphi (l_1) r_{2}^2 + a_{15} \varphi (l_1) f_{sp}^2 l_1^2 \cos (2 \alpha )/4 \nonumber \\&+ a_{15} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&- a_{15} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha )\nonumber \\&- 2 a_{15} \varphi (l_1)^2 r_{1} r_{2}; \end{aligned}$$
(B.29)
$$\begin{aligned}&\mathbf {\psi }_{56}=-a_{15} c_{1}^2/2 - a_{15} d_{1}^2/2 \nonumber \\&- a_{15} r_{2}^2 - a_{16} - a_{18} \nonumber \\&+ a_{15} \varphi (l_1) a_{1} c_{1} \nonumber \\&- a_{15} f_{sp}^2 l_1^2 \cos (2 \alpha )/4 \nonumber \\&+ a_{15} \varphi (l_1) b_{1} d_{1} \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1}^2/2 \nonumber \\&- a_{15} \varphi (l_1)^2 b_{1}^2/2 \nonumber \\&- a_{15} f_{sp}^2 l_1^2/4 - a_{15} \varphi (l_1)^2 r_{1}^2 \nonumber \\&+ a_{15} f_{sp} l_1 d_{1} \cos (\alpha ) - a_{15} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&+ 2 a_{15} \varphi (l_1) r_{1} r_{2}; \end{aligned}$$
(B.30)
$$\begin{aligned}&\mathbf {\psi }_{61}=-a_{1} \varphi (l_1)^3 r_{1} \xi _{n} \nonumber \\&+ a_{1} \varphi (l_1)^2 r_{2} \xi _{n} + c_{1} \varphi (l_1)^2 r_{1} \xi _{n} \nonumber \\&- c_{1} \varphi (l_1) r_{2} \xi _{n}; \end{aligned}$$
(B.31)
$$\begin{aligned}&\mathbf {\psi }_{62}=\xi _{n} \varphi (l_1) f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&- \xi _{n} \varphi (l_1)^2 r_{1} f_{sp} l_1 \cos (\alpha ) + \xi _{n} \varphi (l_1)^2 r_{1} d_{1} \nonumber \\&- \xi _{n} \varphi (l_1)^3 r_{1} b_{1} \nonumber \\&- \xi _{n} \varphi (l_1) r_{2} d_{1} + \xi _{n} \varphi (l_1)^2 b_{1} r_{2}; \end{aligned}$$
(B.32)
$$\begin{aligned}&\mathbf {\psi }_{63}=a_{1} \varphi (l_1)^2 r_{1} \xi _{n}\nonumber \\&- a_{1} \varphi (l_1) r_{2} \xi _{n} - c_{1} \varphi (l_1) r_{1} \xi _{n} \nonumber \\&+ c_{1} r_{2} \xi _{n}; \end{aligned}$$
(B.33)
$$\begin{aligned}&\mathbf {\psi }_{64}=-\xi _{n} f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&+ \xi _{n} r_{2} d_{1} \nonumber \\&+ \xi _{n} \varphi (l_1) r_{1} f_{sp} l_1 \cos (\alpha ) - \xi _{n} \varphi (l_1) r_{1} d_{1} \nonumber \\&+ \xi _{n} \varphi (l_1)^2 r_{1} b_{1} \nonumber \\&- \xi _{n} \varphi (l_1) b_{1} r_{2}; \end{aligned}$$
(B.34)
$$\begin{aligned}&\mathbf {\psi }_{65}=-\xi _{n} \varphi (l_1) f_{sp}^2 l_1^2 \cos (2 \alpha )/4 \nonumber \\&- g_v \varphi (l_1) - \xi _{nl} \varphi (l_1) + \xi _{n} \varphi (l_1) f_{sp} l_1 d_{1} \cos (\alpha ) \nonumber \\&- \xi _{n} \varphi (l_1)^2 b_{1} f_{sp} l_1 \cos (\alpha ) - \xi _{n} \varphi (l_1) c_{1}^2/2 \nonumber \\&- \xi _{n} \varphi (l_1) d_{1}^2/2 - \xi _{n} \varphi (l_1)^3 a_{1}^2/2 \nonumber \\&- \xi _{n} \varphi (l_1)^3 b_{1}^2/2 - \xi _{n} \varphi (l_1)^3 r_{1}^2 \nonumber \\&- \xi _{n} \varphi (l_1) r_{2}^2 + \xi _{n} \varphi (l_1)^2 a_{1} c_{1} + \xi _{n} \varphi (l_1)^2 b_{1} d_{1} \nonumber \\&- \xi _{n} \varphi (l_1) f_{sp}^2 l_1^2/4 + 2 \xi _{n} \varphi (l_1)^2 r_{1} r_{2}; \end{aligned}$$
(B.35)
$$\begin{aligned}&\mathbf {\psi }_{66}=-\xi _{n} f_{sp} l_1 d_{1} \cos (\alpha ) + \xi _{n} c_{1}^2/2 \nonumber \\&+ \xi _{n} d_{1}^2/2 + \xi _{n} r_{2}^2 + \xi _{n} \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&+ g_v + \xi _{nl} \nonumber \\&+ \xi _{n} \varphi (l_1)^2 a_{1}^2/2 + \xi _{n} \varphi (l_1)^2 b_{1}^2/2 + \xi _{n} f_{sp}^2 l_1^2/4 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 r_{1}^2 + \xi _{n} f_{sp}^2 l_1^2 \cos (2 \alpha )/4 \nonumber \\&- \xi _{n} \varphi (l_1) a_{1} c_{1} - \xi _{n} \varphi (l_1) b_{1} d_{1} \nonumber \\&- 2 \xi _{n} \varphi (l_1) r_{1} r_{2}; \end{aligned}$$
(B.36)
$$\begin{aligned}&\mathbf {\Sigma }_{1}=\mu b_{1} \varpi - a_{16} d_{1} \varpi + 2 a_{12} r_{1} a_{1} \nonumber \\&+ 3 a_{13} r_{1}^2 a_{1} - 3 a_{14} r_{2}^2 c_{1} + a_{18} \varphi (l_1) a_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&+ a_{18} \varphi (l_1) b_{1} \varpi \cos (\varpi \gamma ) - (9 a_{14} \varphi (l_1)^2 a_{1}^2 c_{1})/4 \nonumber \\&- (3 a_{14} \varphi (l_1)^2 b_{1}^2 c_{1})/4 \nonumber \\&+ (3 a_{14} \varphi (l_1)^2 a_{1} b_{1} f_{sp} l_1 \cos (\alpha ))/2 + (9 a_{14} \varphi (l_1) a_{1} c_{1}^2)/4 \nonumber \\&- (3 a_{14} f_{sp}^2 l_1^2 c_{1})/8+ (3 a_{15} \varphi (l_1)^2 b_{1}^2 \varpi f_{sp} l_1 \cos (\alpha ))/4 \nonumber \\&- (3 a_{14} f_{sp}^2 l_1^2 c_{1} \cos (2 \alpha ))/8 \nonumber \\&- (3 a_{14} \varphi (l_1)^2 a_{1} b_{1} d_{1})/2\nonumber \\&+ (3 a_{15} \varphi (l_1) b_{1} \varpi d_{1}^2)/4 \nonumber \\&- a_{18} c_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&- a_{18} d_{1} \varpi \cos (\varpi \gamma ) - 3 a_{14} \varphi (l_1)^2 r_{1}^2 c_{1} \nonumber \\&+ 3 a_{14} \varphi (l_1)^3 r_{1}^2 a_{1} - a_{15} d_{1} \varpi r_{2}^2 + 3 a_{14} \varphi (l_1) a_{1} r_{2}^2 \nonumber \\&+ a_{16} \varphi (l_1) b_{1} \varpi \nonumber \\&+ (3 a_{15} f_{sp} l_1 \varpi d_{1}^2 \cos (\alpha ))/4 \nonumber \\&+ a_{18} f_{sp} l_1 \varpi (\cos (\varpi \gamma ) \cos (\alpha )+ \sin (\varpi \gamma ) \sin (\alpha ))/2 \nonumber \\&+ a_{16} f_{sp} l_1 \varpi \cos (\alpha ) - a_{15} c_{1}^2 \varpi d_{1}/4 \nonumber \\&+ a_{15} f_{sp}^3 l_1^3 \varpi \cos (3 \alpha )/16- 2 a_{15} f_{sp} l_1 \varpi \varphi (l_1) r_{1} r_{2} \cos (\alpha ) \nonumber \\&- (3 a_{15} f_{sp}^2 l_1^2 \varpi d_{1})/8 + a_{15} \varphi (l_1) a_{1} \varpi c_{1} d_{1}/2 \nonumber \\&+ a_{18} f_{sp} l_1 \varpi (\cos (\varpi \gamma ) \cos (\alpha ) \nonumber \\&- \sin (\varpi \gamma ) \sin (\alpha ))/2 \nonumber \\&- a_{15} d_{1} \varpi \varphi (l_1)^2 a_{1}^2/4 - (3 a_{15} \varphi (l_1) b_{1} \varpi f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ a_{15} f_{sp} l_1 \varpi \varphi (l_1)^2 a_{1}^2 \cos (\alpha )/4 \nonumber \\&- (3 a_{14} c_{1}^3)/4 + a_{19} + a_{15} f_{sp} l_1 \varpi \varphi (l_1)^2 r_{1}^2 \cos (\alpha ) \nonumber \\&- 6 a_{14} \varphi (l_1)^2 r_{1} a_{1} r_{2} - a_{15} d_{1} \varpi \varphi (l_1)^2 r_{1}^2\nonumber \\&+ a_{15} \varphi (l_1) b_{1} \varpi r_{2}^2 + 6 a_{14} \varphi (l_1) r_{1} r_{2} c_{1} \nonumber \\&+ a_{15} \varphi (l_1)^3 b_{1} \varpi r_{1}^2 + (3 a_{14} \varphi (l_1)^3 a_{1}^3)/4 \nonumber \\&+ (3 a_{15} \varphi (l_1) b_{1} \varpi f_{sp}^2 l_1^2)/8 \nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} \varpi b_{1} c_{1}/2\nonumber \\&- (3 a_{15} f_{sp}^2 l_1^2 \varpi d_{1} \cos (2 \alpha ))/8 \nonumber \\&+ (3 a_{14} \varphi (l_1) a_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 + a_{15} \varphi (l_1)^3 b_{1}^3 \varpi /4 \nonumber \\&+ (3 a_{15} \varphi (l_1) b_{1} \varpi f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- (3 a_{14} \varphi (l_1) b_{1} f_{sp} l_1 c_{1} \cos (\alpha ))/2 \nonumber \\&+ (3 a_{15} f_{sp}^3 l_1^3 \varpi \cos (\alpha ))/16 + 2 a_{15} d_{1} \varpi \varphi (l_1) r_{1} r_{2} \nonumber \\&+ a_{15} f_{sp} l_1 \varpi r_{2}^2 \cos (\alpha ) \nonumber \\&- 2 a_{15} \varphi (l_1)^2 b_{1} \varpi r_{1} r_{2} + (3 a_{13} a_{1} b_{1}^2)/4 \nonumber \\&- (3 a_{14} c_{1} d_{1}^2)/4 + a_{11} a_{1} \nonumber \\&+ b_{12} a_{1} - a_{1} \varpi ^2 + a_{15} f_{sp} l_1 \varpi c_{1}^2 \cos (\alpha )/4 \nonumber \\&- (3 a_{14} \varphi (l_1) a_{1} f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ (3 a_{14} \varphi (l_1) a_{1} f_{sp}^2 l_1^2)/8 + a_{15} \varphi (l_1)^3 b_{1} \varpi a_{1}^2/4 \nonumber \\&+ a_{15} \varphi (l_1) b_{1} \varpi c_{1}^2/4 + (3 a_{14} f_{sp} l_1 c_{1} d_{1} \cos (\alpha ))/2 \nonumber \\&- (3 a_{15} \varphi (l_1)^2 b_{1}^2 \varpi d_{1})/4 + (3 a_{14} \varphi (l_1) b_{1} c_{1} d_{1})/2\nonumber \\&- a_{15} d_{1}^3 \varpi /4 + (3 a_{14} \varphi (l_1)^3 a_{1} b_{1}^2)/4\nonumber \\&+ (3 a_{13} a_{1}^3)/4 - a_{15} f_{sp} l_1 \varpi \varphi (l_1) a_{1} c_{1} \cos (\alpha )/2 \nonumber \\&- (3 b_{11} a_{1})/4 + (3 a_{14} \varphi (l_1) a_{1} d_{1}^2)/4; \end{aligned}$$
(B.37)
$$\begin{aligned}&\mathbf {\Sigma }_{2}=(3 a_{14} f_{sp} l_1 c_{1}^2 \cos (\alpha ))/4 \nonumber \\&- \mu a_{1} \varpi + a_{16} c_{1} \varpi + 3 a_{13} r_{1}^2 b_{1} \nonumber \\&+ 2 a_{12} r_{1} b_{1} - 3 a_{14} r_{2}^2 d_{1} \nonumber \\&+ a_{15} \varphi (l_1)^2 b_{1}^2 \varpi c_{1}/4 - (3 a_{14} \varphi (l_1)^2 a_{1} b_{1} c_{1})/2 \nonumber \\&- (9 a_{14} f_{sp}^2 l_1^2 d_{1} \cos (2 \alpha ))/8 \nonumber \\&+ (3 a_{15} \varphi (l_1)^2 a_{1}^2 \varpi c_{1})/4 + a_{15} f_{sp}^2 l_1^2 \varpi c_{1}/8 \nonumber \\&- (3 a_{15} \varphi (l_1) a_{1} \varpi c_{1}^2)/4 + (9 a_{14} f_{sp} l_1 d_{1}^2 \cos (\alpha ))/4 \nonumber \\&- a_{15} \varphi (l_1) b_{1} \varpi c_{1} d_{1}/2 + (3 a_{14} \varphi (l_1) a_{1} c_{1} d_{1})/2 \nonumber \\&- a_{15} \varphi (l_1) a_{1} \varpi d_{1}^2/4 \nonumber \\&- (9 a_{14} \varphi (l_1) b_{1} f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&- (3 a_{14} \varphi (l_1) a_{1} f_{sp} l_1 c_{1} \cos (\alpha ))/2 \nonumber \\&- a_{15} \varphi (l_1) a_{1} \varpi f_{sp}^2 l_1^2 \cos (2 \alpha )/8 \nonumber \\&+ 3 a_{14} \varphi (l_1)^3 r_{1}^2 b_{1} - 3 a_{14} \varphi (l_1)^2 r_{1}^2 d_{1} \nonumber \\&+ 3 a_{14} \varphi (l_1) b_{1} r_{2}^2 \nonumber \\ &+ a_{15} c_{1} \varpi r_{2}^2 - a_{16} \varphi (l_1) a_{1} \varpi + a_{18} c_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&- a_{18} d_{1} \varpi \sin (\varpi \gamma ) - a_{15} \varphi (l_1)^3 b_{1}^2 \varpi a_{1}/4 \nonumber \\&+ (9 a_{14} \varphi (l_1) b_{1} f_{sp}^2 l_1^2)/8 + a_{15} \varphi (l_1) a_{1} \varpi f_{sp} l_1 d_{1} \cos (\alpha )/2 \nonumber \\&+ (9 a_{14} \varphi (l_1)^2 b_{1}^2 f_{sp} l_1 \cos (\alpha ))/4 \nonumber \\&+ (9 a_{14} \varphi (l_1) b_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- a_{15} \varphi (l_1) a_{1} \varpi f_{sp}^2 l_1^2/8 + (3 a_{14} \varphi (l_1)^2 a_{1}^2 f_{sp} l_1 \cos (\alpha ))/4 \nonumber \\&+ a_{18} f_{sp} l_1 \varpi (\sin (\varpi \gamma ) \cos (\alpha ) - \cos (\varpi \gamma ) \sin (\alpha ))/2 \nonumber \\&+ a_{18} f_{sp} l_1 \varpi (\sin (\varpi \gamma ) \cos (\alpha ) + \cos (\varpi \gamma ) \sin (\alpha ))/2\nonumber \\&+ (3 a_{14} f_{sp}^3 l_1^3 \cos (3 \alpha ))/16 + (9 a_{14} f_{sp}^3 l_1^3 \cos (\alpha ))/16 \nonumber \\&- (9 a_{14} f_{sp}^2 l_1^2 d_{1})/8 - (9 a_{14} \varphi (l_1)^2 b_{1}^2 d_{1})/4 \nonumber \\&+ a_{15} d_{1}^2 \varpi c_{1}/4 - (3 a_{14} \varphi (l_1)^2 a_{1}^2 d_{1})/4 \nonumber \\&+ (9 a_{14} \varphi (l_1) b_{1} d_{1}^2)/4 \nonumber \\&+ (3 a_{14} \varphi (l_1) b_{1} c_{1}^2)/4 \nonumber \\&- a_{15} \varphi (l_1)^3 a_{1}^3 \varpi /4 \nonumber \\&+ (3 a_{14} \varphi (l_1)^3 a_{1}^2 b_{1})/4 \nonumber \\&+ a_{15} c_{1} \varpi \varphi (l_1) b_{1} f_{sp} l_1 \cos (\alpha )/2\nonumber \\&- a_{15} \varphi (l_1)^2 a_{1} \varpi b_{1} f_{sp} l_1 \cos (\alpha )/2 \nonumber \\&+ a_{15} c_{1}^3 \varpi /4 - (3 a_{14} c_{1}^2 d_{1})/4 \nonumber \\&+ (3 a_{13} a_{1}^2 b_{1})/4 \nonumber \\&+ (3 a_{14} \varphi (l_1)^3 b_{1}^3)/4 - (3 a_{14} d_{1}^3)/4 - b_{11} b_{1}/4 \nonumber \\&+ (3 a_{13} b_{1}^3)/4 - a_{15} f_{sp} l_1 \varpi c_{1} d_{1} \cos (\alpha )/2\nonumber \\&+ a_{15} c_{1} \varpi \varphi (l_1)^2 r_{1}^2 + 3 a_{14} f_{sp} l_1 r_{2}^2 \cos (\alpha ) \nonumber \\&+ 6 a_{14} \varphi (l_1) r_{1} r_{2} d_{1} - a_{15} \varphi (l_1) a_{1} \varpi r_{2}^2 \nonumber \\&- 6 a_{14} \varphi (l_1)^2 r_{1} b_{1} r_{2} + a_{20} - a_{18} \varphi (l_1) a_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&+ a_{18} \varphi (l_1) b_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&- a_{15} \varphi (l_1)^3 a_{1} \varpi r_{1}^2\nonumber \\&+ a_{15} \varphi (l_1)^2 b_{1} \varpi a_{1} d_{1}/2 \nonumber \\&+ a_{15} f_{sp}^2 l_1^2 \varpi c_{1} \cos (2 \alpha )/8 \nonumber \\&- 6 a_{14} \varphi (l_1) r_{1} f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&+ a_{11} b_{1} - b_{1} \varpi ^2 + b_{12} b_{1} + 3 a_{14} \varphi (l_1)^2 r_{1}^2 f_{sp} l_1 \cos (\alpha ) \nonumber \\&+ 2 a_{15} \varphi (l_1)^2 a_{1} \varpi r_{1} r_{2} \nonumber \\&- 2 a_{15} c_{1} \varpi \varphi (l_1) r_{1} r_{2}; \end{aligned}$$
(B.38)
$$\begin{aligned}&\mathbf {\Sigma }_{3}=3 k_n r_{2}^2 c_{1} + \xi _{nl} d_{1} \varpi \nonumber \\&- (3 k_n \varphi (l_1) a_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- (3 \xi _{n} f_{sp}^3 l_1^3 \varpi \cos (\alpha ))/16 - c_{1} \varpi ^2\nonumber \\&- \xi _{n} c_{1}^2 \varpi f_{sp} l_1 \cos (\alpha )/4 \nonumber \\&- (3 \xi _{n} \varphi (l_1)^2 b_{1}^2 \varpi f_{sp} l_1 \cos (\alpha ))/4 \nonumber \\&- \xi _{nl} \varphi (l_1) b_{1} \varpi \nonumber \\&+ g_v c_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&+ Gv d_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&- 3 k_n \varphi (l_1)^3 r_{1}^2 a_{1} \nonumber \\&+ 3 k_n \varphi (l_1)^2 r_{1}^2 c_{1} \nonumber \\&- 3 k_n \varphi (l_1) a_{1} r_{2}^2 \nonumber \\&+ \xi _{n} d_{1} \varpi r_{2}^2 \nonumber \\&+ (9 k_n \varphi (l_1)^2 a_{1}^2 c_{1})/4 + (3 k_n \varphi (l_1)^2 b_{1}^2 c_{1})/4 \nonumber \\&- \xi _{n} c_{1}^2 \varpi \varphi (l_1) b_{1}/4 \nonumber \\&- (3 \xi _{n} \varphi (l_1) b_{1} \varpi f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&+ (3 k_n \varphi (l_1) a_{1} f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ \xi _{n} d_{1} \varpi \varphi (l_1)^2 r_{1}^2 \nonumber \\&- 6 k_n \varphi (l_1) r_{1} r_{2} c_{1} - \xi _{n} \varphi (l_1) b_{1} \varpi r_{2}^2 \nonumber \\&+ 6 k_n \varphi (l_1)^2 r_{1} a_{1} r_{2} \nonumber \\&- g_v \varphi (l_1) a_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&- g_v \varphi (l_1) b_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&- \xi _{n} \varphi (l_1)^3 b_{1} \varpi r_{1}^2 \nonumber \\&+ 2 \xi _{n} f_{sp} l_1 \varpi \varphi (l_1) r_{1} r_{2} \cos (\alpha )\nonumber \\&- \xi _{n} \varphi (l_1)^3 b_{1}^3 \varpi /4 \nonumber \\&- (9 k_n \varphi (l_1) a_{1} c_{1}^2)/4 - \xi _{nl} f_{sp} l_1 \varpi \cos (\alpha ) \nonumber \\&+ (3 k_n f_{sp}^2 l_1^2 c_{1} \cos (2 \alpha ))/8 \nonumber \\&- g_v f_{sp} l_1 \varpi (\cos (\varpi \gamma ) \cos (\alpha ) \nonumber \\&+ \sin (\varpi \gamma ) \sin (\alpha ))/2 - \xi _{n} f_{sp} l_1 \varpi \varphi (l_1)^2 r_{1}^2 \cos (\alpha ) \nonumber \\&+ (3 \xi _{n} f_{sp}^2 l_1^2 \varpi d_{1})/8 - \xi _{n} \nonumber \\&\varphi (l_1)^3 a_{1}^2 \varpi b_{1}/4 \nonumber \\&- \xi _{n} f_{sp} l_1 \varpi r_{2}^2 \cos (\alpha ) + 2 \xi _{n} \varphi (l_1)^2 b_{1} \varpi r_{1} r_{2} \nonumber \\&- 2 \xi _{n} d_{1} \varpi \varphi (l_1) r_{1} r_{2} + \xi _{n} \varphi (l_1)^2 a_{1}^2 \varpi d_{1}/4 \nonumber \\&- (3 k_n \varphi (l_1) a_{1} f_{sp}^2 l_1^2)/8 \nonumber \\&+ (3 k_n \varphi (l_1) b_{1} f_{sp} l_1 c_{1} \cos (\alpha ))/2 + (3 k_n \varphi (l_1)^2 a_{1} b_{1} d_{1})/2 \nonumber \\&- (3 \xi _{n} \varphi (l_1) b_{1} \varpi d_{1}^2)/4 + (3 k_n f_{sp}^2 l_1^2 c_{1})/8 \nonumber \\&- \xi _{n} f_{sp}^3 l_1^3 \varpi \cos (3 \alpha )/16 \nonumber \\&- (3 k_n \varphi (l_1) a_{1} d_{1}^2)/4 + \xi _{n} d_{1} \varpi c_{1}^2/4 \nonumber \\&+ (3 k_n c_{1}^3)/4 + (3 \xi _{n} \varphi (l_1) b_{1} \varpi f_{sp} l_1 d_{1} \cos (\alpha ))/2 \nonumber \\&+ \xi _{n} \varphi (l_1) a_{1} \varpi f_{sp} l_1 c_{1} \cos (\alpha )/2\nonumber \\&- (3 k_n \varphi (l_1)^3 a_{1} b_{1}^2)/4 + \xi _{n} \varphi (l_1)^2 a_{1} \varpi b_{1} c_{1}/2 \nonumber \\&+ (3 k_n c_{1} d_{1}^2)/4 + \xi _{n} d_{1}^3 \varpi /4 - (3 k_n \varphi (l_1)^3 a_{1}^3)/4 \nonumber \\&- (3 k_n \varphi (l_1)^2 a_{1} b_{1} f_{sp} l_1 \cos (\alpha ))/2 \nonumber \\&+ (3 \xi _{n} f_{sp}^2 l_1^2 \varpi d_{1} \cos (2 \alpha ))/8 \nonumber \\&- (3 k_n f_{sp} l_1 c_{1} d_{1} \cos (\alpha ))/2 \nonumber \\&+ (3 \xi _{n} \varphi (l_1)^2 b_{1}^2 \varpi d_{1})/4 - \xi _{n} f_{sp} l_1 \varpi \varphi (l_1)^2 a_{1}^2 \cos (\alpha )/4 \nonumber \\&- (3 \xi _{n} f_{sp} l_1 \varpi d_{1}^2 \cos (\alpha ))/4 \nonumber \\&- g_v f_{sp} l_1 \varpi (\cos (\varpi \gamma ) \cos (\alpha ) \nonumber \\&- \sin (\varpi \gamma ) \sin (\alpha ))/2 \nonumber \\&- \xi _{n} \varphi (l_1) a_{1} \varpi c_{1} d_{1}/2 \nonumber \\&- (3 k_n \varphi (l_1) b_{1} c_{1} d_{1})/2 \nonumber \\&- (3 \xi _{n} \varphi (l_1) b_{1} \varpi f_{sp}^2 l_1^2)/8; \end{aligned}$$
(B.39)
$$\begin{aligned}&\mathbf {\Sigma }_{4}=-\xi _{nl} c_{1} \varpi + 3 k_n r_{2}^2 d_{1} \nonumber \\&- \xi _{n} f_{sp} l_1 \varpi \varphi (l_1) a_{1} d_{1} \cos (\alpha )/2 \nonumber \\&- d_{1} \varpi ^2 - g_v f_{sp} l_1 \varpi (\sin (\varpi \gamma ) \cos (\alpha )\nonumber \\&+ \cos (\varpi \gamma ) \sin (\alpha ))/2 - 3 k_n \varphi (l_1)^3 r_{1}^2 b_{1} - 3 k_n \varphi (l_1) b_{1} r_{2}^2\nonumber \\&- \xi _{n} c_{1} \varpi r_{2}^2 + g_v d_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&+ 3 k_n \varphi (l_1)^2 r_{1}^2 d_{1} + \xi _{nl} \varphi (l_1) a_{1} \varpi - g_v c_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&- \xi _{n} \varphi (l_1) b_{1} \varpi f_{sp} l_1 c_{1} \cos (\alpha )/2\nonumber \\&+ (9 k_n f_{sp}^2 l_1^2 d_{1} \cos (2 \alpha ))/8 \nonumber \\&- (9 k_n \varphi (l_1) b_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/8 \nonumber \\&- \xi _{n} c_{1} \varpi \varphi (l_1)^2 r_{1}^2 \nonumber \\&+ \xi _{n} \varphi (l_1)^3 a_{1} \varpi r_{1}^2 - 6 k_n \varphi (l_1) r_{1} r_{2} d_{1} \nonumber \\&+ 6 k_n \varphi (l_1)^2 r_{1} b_{1} r_{2} + \xi _{n} \varphi (l_1) a_{1} \varpi r_{2}^2 \nonumber \\&- 3 k_n f_{sp} l_1 r_{2}^2 \cos (\alpha ) \nonumber \\&+ g_v \varphi (l_1) a_{1} \varpi \cos (\varpi \gamma ) \nonumber \\&- g_v \varphi (l_1) b_{1} \varpi \sin (\varpi \gamma ) \nonumber \\&- g_v f_{sp} l_1 \varpi (\sin (\varpi \gamma ) \cos (\alpha ) - \cos (\varpi \gamma ) \sin (\alpha ))/2 \nonumber \\&- \xi _{n} \varphi (l_1)^2 b_{1} \varpi a_{1} d_{1}/2 \nonumber \\&+ \xi _{n} d_{1}^2 \varpi \varphi (l_1) a_{1}/4 - \xi _{n} f_{sp}^2 l_1^2 \varpi c_{1}/8 \nonumber \\&- (9 k_n f_{sp}^3 l_1^3 \cos (\alpha ))/16 + \xi _{n} \varphi (l_1) b_{1} \varpi c_{1} d_{1}/2 \nonumber \\&- (9 k_n \varphi (l_1)^2 b_{1}^2 f_{sp} l_1 \cos (\alpha ))/4 + 6 k_n \varphi (l_1) r_{1} f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&- (3 k_n \varphi (l_1)^2 a_{1}^2 f_{sp} l_1 \cos (\alpha ))/4 \nonumber \\&- (3 k_n \varphi (l_1)^3 a_{1}^2 b_{1})/4 \nonumber \\&- (3 k_n \varphi (l_1) b_{1} c_{1}^2)/4 \nonumber \\&+ \xi _{n} \varphi (l_1)^2 b_{1} \varpi a_{1} f_{sp} l_1 \cos (\alpha )/2 \nonumber \\&+ (9 k_n \varphi (l_1) b_{1} f_{sp} l_1 d_{1} \cos (\alpha ))/2 - (3 k_n f_{sp} l_1 c_{1}^2 \cos (\alpha ))/4 \nonumber \\&- 3 k_n \varphi (l_1)^2 r_{1}^2 f_{sp} l_1 \cos (\alpha ) \nonumber \\&- 2 \xi _{n} \varphi (l_1)^2 a_{1} \varpi r_{1} r_{2} \nonumber \\&+ 2 \xi _{n} c_{1} \varpi \varphi (l_1) r_{1} r_{2} - \xi _{n} c_{1}^3 \varpi /4 + (3 k_n d_{1}^3)/4 \nonumber \\&+ (3 \xi _{n} \varphi (l_1) a_{1} \varpi c_{1}^2)/4 \nonumber \\&+ (3 k_n \varphi (l_1)^2 a_{1} b_{1} c_{1})/2 \nonumber \\&- (3 \xi _{n} \varphi (l_1)^2 a_{1}^2 \varpi c_{1})/4 \nonumber \\&+ \xi _{n} \varphi (l_1) a_{1} \varpi f_{sp}^2 l_1^2/8 \nonumber \\&- (3 k_n \varphi (l_1) a_{1} c_{1} d_{1})/2 \nonumber \\&+ (3 k_n \varphi (l_1) a_{1} f_{sp} l_1 c_{1} \cos (\alpha ))/2 \nonumber \\&+ \xi _{n} \varphi (l_1) a_{1} \varpi f_{sp}^2 l_1^2 \cos (2 \alpha )/8 \nonumber \\&+ \xi _{n} c_{1} \varpi f_{sp} l_1 d_{1} \cos (\alpha )/2 \nonumber \\&- (9 k_n \varphi (l_1) b_{1} d_{1}^2)/4 + (9 k_n \varphi (l_1)^2 b_{1}^2 d_{1})/4 \nonumber \\&- \xi _{n} f_{sp}^2 l_1^2 \varpi c_{1} \cos (2 \alpha )/8 + \xi _{n} \varphi (l_1)^3 a_{1}^3 \varpi /4 \nonumber \\&+ \xi _{n} \varphi (l_1)^3 b_{1}^2 \varpi a_{1}/4 - \xi _{n} \varphi (l_1)^2 b_{1}^2 \varpi c_{1}/4 \nonumber \\&- \xi _{n} c_{1} \varpi d_{1}^2/4 + (3 k_n \varphi (l_1)^2 a_{1}^2 d_{1})/4 \nonumber \\&- (3 k_n \varphi (l_1)^3 b_{1}^3)/4 \nonumber \\&+ (3 k_n c_{1}^2 d_{1})/4 + (9 k_n f_{sp}^2 l_1^2 d_{1})/8\nonumber \\&- (9 k_n f_{sp} l_1 d_{1}^2 \cos (\alpha ))/4 \nonumber \\&- (3 k_n f_{sp}^3 l_1^3 \cos (3 \alpha ))/16 \nonumber \\&- (9 k_n \varphi (l_1) b_{1} f_{sp}^2 l_1^2)/8; \end{aligned}$$
(B.40)
$$\begin{aligned}&\mathbf {\Sigma }_{5}=3 a_{14} f_{sp} l_1 r_{2} d_{1} \cos (\alpha ) \nonumber \\&+ a_{14} \varphi (l_1)^3 r_{1}^3 \nonumber \\&+ (3 a_{14} \varphi (l_1) r_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/4 \nonumber \\&- (3 a_{14} \varphi (l_1)^2 b_{1}^2 r_{2})/2 \nonumber \\&- (3 a_{14} f_{sp}^2 l_1^2 r_{2})/4 + 3 a_{14} \varphi (l_1) r_{1} r_{2}^2 \nonumber \\&- 3 a_{14} \varphi (l_1)^2 r_{1}^2 r_{2} - 3 a_{14} \varphi (l_1)^2 r_{1} a_{1} c_{1} \nonumber \\&- 3 a_{14} \varphi (l_1)^2 r_{1} b_{1} d_{1} \nonumber \\&+ 3 a_{14} \varphi (l_1) a_{1} r_{2} c_{1} + 3 a_{14} \varphi (l_1) b_{1} r_{2} d_{1} \nonumber \\&+ a_{17}/2 + a_{12} b_{1}^2/2 + a_{21} - (3 a_{14} r_{2} c_{1}^2)/2 \nonumber \\&+ (3 a_{13} r_{1} b_{1}^2)/2 \nonumber \\&+ (3 a_{13} r_{1} a_{1}^2)/2 - (3 a_{14} r_{2} d_{1}^2)/2 + a_{12} a_{1}^2/2 \nonumber \\&+ (3 a_{14} \varphi (l_1)^3 r_{1} a_{1}^2)/2 + (3 a_{14} \varphi (l_1)^3 r_{1} b_{1}^2)/2 \nonumber \\&+ (3 a_{14} \varphi (l_1) r_{1} c_{1}^2)/2 \nonumber \\&+ a_{11} r_{1} + b_{12} r_{1} \nonumber \\&+ a_{12} r_{1}^2 + a_{13} r_{1}^3 - a_{14} r_{2}^3 - (3 a_{14} f_{sp}^2 l_1^2 r_{2} \cos (2 \alpha ))/4 \nonumber \\&+ 3 a_{14} \varphi (l_1)^2 r_{1} b_{1} f_{sp} l_1 \cos (\alpha ) - 3 a_{14} \varphi (l_1) r_{1} f_{sp} l_1 d_{1} \cos (\alpha ) \nonumber \\&- 3 a_{14} \varphi (l_1) b_{1} f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&+ (3 a_{14} \varphi (l_1) r_{1} d_{1}^2)/2 - (3 a_{14} \varphi (l_1)^2 a_{1}^2 r_{2})/2 \nonumber \\&+ (3 a_{14} \varphi (l_1) r_{1} f_{sp}^2 l_1^2)/4 - b_{11} r_{1}/2; \end{aligned}$$
(B.41)
$$\begin{aligned}&\mathbf {\Sigma }_{6}=-3 k_n f_{sp} l_1 r_{2} d_{1} \cos (\alpha ) \nonumber \\&- k_n \varphi (l_1)^3 r_{1}^3 - (3 k_n \varphi (l_1) r_{1} f_{sp}^2 l_1^2 \cos (2 \alpha ))/4 + k_n r_{2}^3 \nonumber \\&+ (3 k_n r_{2} d_{1}^2)/2 + (3 k_n r_{2} c_{1}^2)/2 - (3 k_n \varphi (l_1)^3 r_{1} a_{1}^2)/2 \nonumber \\&+ (3 k_n \varphi (l_1)^2 a_{1}^2 r_{2})/2 \nonumber \\&- (3 k_n \varphi (l_1)^3 r_{1} b_{1}^2)/2 \nonumber \\&+ (3 k_n \varphi (l_1)^2 b_{1}^2 r_{2})/2 \nonumber \\&- (3 k_n \varphi (l_1) r_{1} c_{1}^2)/2 - (3 k_n \varphi (l_1) r_{1} d_{1}^2)/2\nonumber \\&+ 3 k_n \varphi (l_1)^2 r_{1}^2 r_{2} \nonumber \\&- 3 k_n \varphi (l_1) r_{1} r_{2}^2 + (3 k_n f_{sp}^2 l_1^2 r_{2})/4 + 3 k_n \varphi (l_1)^2 r_{1} a_{1} c_{1} \nonumber \\&- 3 k_n \varphi (l_1) a_{1} r_{2} c_{1} \nonumber \\&+ 3 k_n \varphi (l_1)^2 r_{1} b_{1} d_{1} - 3 k_n \varphi (l_1) b_{1} r_{2} d_{1} \nonumber \\&- (3 k_n \varphi (l_1) r_{1} f_{sp}^2 l_1^2)/4 + (3 k_n f_{sp}^2 l_1^2 r_{2} \cos (2 \alpha ))/4 \nonumber \\&+ 3 k_n \varphi (l_1) b_{1} f_{sp} l_1 r_{2} \cos (\alpha ) \nonumber \\&- 3 k_n \varphi (l_1)^2 r_{1} b_{1} f_{sp} l_1 \cos (\alpha ) \nonumber \\&+ 3 k_n \varphi (l_1) r_{1} f_{sp} l_1 d_{1} \cos (\alpha ); \end{aligned}$$
(B.42)

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Liu, Z., Ding, W. Vibration suppression of a stay cable using a time-delayed nonlinear energy sink under multi-source external excitation. Nonlinear Dyn 114, 30 (2026). https://doi.org/10.1007/s11071-025-11913-7

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