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Nonlinear dynamic analysis of a cable-stayed beam with a nonlinear energy sink

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Abstract

In recent years, nonlinear energy sink (NES) has received widespread attention from scholars as an efficient passive control means. In this paper, the effect of NES on the nonlinear dynamic response of the cable-beam composite structure is investigated. Firstly, a mechanical model of the cable-beam composite structure with a NES under the external excitation of the beam is established. By using Hamilton principle and Galerkin discretization, the ordinary differential equations (ODEs) of the system are derived. Then, the incremental harmonic balance (IHB) method is employed to obtain the frequency response of the system when the beam is subjected to forced excitation. Finally, three working conditions are considered to discuss the effect of NES on the dynamic characteristics of the composite structure. Moreover, the vibration suppression mechanism of NES attached to the cable on the beam members is also investigated. The results demonstrate that when the natural frequencies of beam and each mode of cable are close, NES has good vibration suppression characteristics for both cable and beam. Furthermore, the vibration mitigation effect of NES on beam members has a great relationship with the degree of cable-beam coupling vibration.

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Acknowledgements

The authors wish to acknowledge the support of the National Natural Science Foundation of China (11972151 and 11872176).

Funding

National Natural Science Foundation of China (11972151, 11872176).

Author information

Authors and Affiliations

  1. Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi University, Nanning, China

    Yifei Wang, Houjun Kang, Yunyue Cong & Tieding Guo

  2. College of Civil Engineering and Architecture, Guangxi University, Nanning, 530004, China

    Yifei Wang, Houjun Kang, Yunyue Cong & Tieding Guo

  3. Scientific Research Center of Engineering Mechanics, Guangxi University, Nanning, 530004, China

    Yifei Wang, Houjun Kang, Yunyue Cong & Tieding Guo

  4. School of Civil Engineering, Nanchang Institute of Technology, Nanchang, 330099, China

    Tao Fu

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  1. Yifei Wang
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Correspondence to Houjun Kang.

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Appendix: A

Appendix: A

The coefficients in Eqs. (11)–(15) are given as:

$$ d_{1} = \int_{0}^{1} {\phi _{1}^{2} (x)} {\text{d}}x;h_{{\text{m}}} = \int_{0}^{1} {\varphi _{{\text{m}}}^{2} (x)} {\text{d}}x,\;\left( {m = 1,2,3} \right); $$
$$ k_{101} = \mu_{{\text{b}}} ; $$
$$k_{{102}} = \int_{0}^{1} {\phi _{1} (x)\phi _{1}^{{(4)}} (x)} dx/\beta _{{\text{b}}}^{4} d_{1} + \kappa _{{\text{c}}} \sin \theta \phi _{1} (x_{1} )\sin \theta \gamma _{{\text{c}}} \phi _{1} (x_{1} )/d_{1} - \kappa _{{\text{c}}} \sin \theta \phi _{1} (x_{1} )\cos \theta \phi _{1} (x_{1} )\int_{0}^{1} {y_{{\text{c}}}^{\prime } (x)} dx/d_{1};$$
$$ k_{103} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{1} {(}x{\text{)d}}x} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{1} {(}x{\text{)d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{104} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{2} {(}x{\text{)d}}x} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{2} {(}x{\text{)d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{105} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{3} {(}x{\text{)d}}x} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}} \varphi^{\prime}_{3} {(}x{\text{)d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{106} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{107} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{1}^{\prime 2} (x)} dx} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{1}^{\prime 2} (x)} dx} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{108} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{2}^{\prime 2} (x)} dx} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{2}^{\prime 2} (x)} dx} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{109} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{3}^{\prime 2} (x)} dx} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{3}^{\prime 2} (x)} dx} {2d_{1} }}} \right. \kern-0pt} {2d_{1} }}; $$
$$ k_{110} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\cos \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{1}^{\prime } (x)} dx} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} (x_{1} )\cos \theta \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{1}^{\prime } (x)} dx} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{111} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{112} = - {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\kappa_{{\text{c}}} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{113} = - {{\eta_{{\text{b}}} \int_{0}^{1} {\phi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\phi_{1} {(}x{)}\phi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\eta_{{\text{b}}} \int_{0}^{1} {\phi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\phi_{1} {(}x{)}\phi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {2\beta_{{\text{b}}}^{4} d_{1} }}} \right. \kern-0pt} {2\beta_{{\text{b}}}^{4} d_{1} }}; $$
$$ k_{114} = {{\int_{0}^{1} {f_{{\text{b}}} \phi_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\int_{0}^{1} {f_{{\text{b}}} \phi_{1} {(}x{)}} {\text{d}}x} {d_{1} }}} \right. \kern-0pt} {d_{1} }}; $$
$$ k_{201} = {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x} {h_{1} }}} \right. \kern-0pt} {h_{1} }}; $$
$$ k_{202} = \mu_{{\text{c}}} ; $$
$$ k_{203} = {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x} {h_{1} }}} \right. \kern-0pt} {h_{1} }}; $$
$$ k_{204} = - {{\left( {\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \right)} \mathord{\left/ {\vphantom {{\left( {\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \right)} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{205} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{206} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{207} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }} + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{208} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{1} }} - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{209} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{210} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{211} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{212} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{213} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ \begin{gathered} k_{214} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}\\\quad\quad - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y^{\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y^{\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }} \hfill \\ \quad \quad + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; \hfill \\ \end{gathered} $$
$$ k_{215} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{216} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$k_{{217}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{1}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{1}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{1} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{218} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{219} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{220} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$ k_{{221}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{2}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{2}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{1} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{1} }};$$
$$ k_{222} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi^{\prime\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{1} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{1} }}; $$
$$k_{{223}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{1} (x)\varphi _{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{1} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{1} }};$$
$$ k_{224} = {{k_{{\text{n}}} \varphi_{1} (l_{1} )} \mathord{\left/ {\vphantom {{k_{{\text{n}}} \varphi_{1} (l_{1} )} {h_{1} }}} \right. \kern-0pt} {h_{1} }}; $$
$$ k_{225} = {{c_{{\text{n}}} \varphi_{1} (l_{1} )} \mathord{\left/ {\vphantom {{c_{{\text{n}}} \varphi_{1} (l_{1} )} {h_{1} }}} \right. \kern-0pt} {h_{1} }}; $$
$$ k_{301} = {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x} {h_{2} }}} \right. \kern-0pt} {h_{2} }}; $$
$$ k_{302} = \mu_{{\text{c}}} ; $$
$$ k_{303} = {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x} {h_{2} }}} \right. \kern-0pt} {h_{2} }}; $$
$$ k_{304} = - {{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{305} = - {{\left( {\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \right)} \mathord{\left/ {\vphantom {{\left( {\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \right)} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{306} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{307} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} {\beta_{c}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{c}^{2} h_{2} }} + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{308} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{309} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{2} }} - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{310} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{311} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{312} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{313} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{314} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ \begin{gathered} k_{315} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }} \\\quad\quad - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y^{\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y^{\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }} \hfill \\ \quad \quad + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; \hfill \\ \end{gathered} $$
$$ k_{316} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{{317}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{2}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{2}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{2} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{318} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{319} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{320} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{{323}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{2} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{322} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi^{\prime\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{2} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{{323}} = - {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} \mathord{\left/ {\vphantom {{\lambda _{{\text{c}}} \int_{0}^{1} {\varphi _{2} (x)\varphi ^{\prime\prime}_{2} (x)} dx\int_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)} dx} {2\beta _{{\text{c}}}^{2} h_{2} }}} \right. \kern-\nulldelimiterspace} {2\beta _{{\text{c}}}^{2} h_{2} }}; $$
$$ k_{324} = {{k_{{\text{n}}} \varphi_{2} (l_{1} )} \mathord{\left/ {\vphantom {{k_{{\text{n}}} \varphi_{2} (l_{1} )} {h_{2} }}} \right. \kern-0pt} {h_{2} }}; $$
$$ k_{325} = {{c_{{\text{n}}} \varphi_{2} (l_{1} )} \mathord{\left/ {\vphantom {{c_{{\text{n}}} \varphi_{2} (l_{1} )} {h_{2} }}} \right. \kern-0pt} {h_{2} }}; $$
$$ k_{401} = {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x} {h_{3} }}} \right. \kern-0pt} {h_{3} }}; $$
$$ k_{402} = \mu_{{\text{c}}} ; $$
$$ k_{403} = {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\mu_{{\text{c}}} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x} {h_{3} }}} \right. \kern-0pt} {h_{3} }}; $$
$$ k_{404} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{1} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{405} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{406} = - {{\left( {\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi^{\prime\prime}_{3} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \right)} \mathord{\left/ {\vphantom {{\left( {\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi^{\prime\prime}_{3} {(}x{)}} {\text{d}}x + \lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \right)} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{407} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }} + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{408} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{1} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{409} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{2} {(}x{)}} {\text{d}}x} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{410} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}y^{\prime\prime}_{{\text{c}}} {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi^{{\prime}{2}}_{3} {(}x{)}} {\text{d}}x} {2\beta_{c}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{c}^{2} h_{3} }} - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi^{\prime\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi^{\prime\prime}_{3} {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{3} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{411} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} (x_{1} )\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \phi_{1}^{2} (x_{1} )\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$k_{412} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {y^{\prime}_{{\text{c}}} (x)\varphi^{\prime}_{1} (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {y^{\prime}_{{\text{c}}} (x)\varphi^{\prime}_{1} (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }};$$
$$ k_{413} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y^{\prime}_{{\text{c}}} {(}x{)}\varphi^{\prime}_{2} {(}x{)}} {\text{d}}x} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{414} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{1} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{1} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{415} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{2} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{2} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ \begin{gathered} k_{416} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{3} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{3} (x)} dx\int_{0}^{1} {\varphi_{3} (x)y_{{\text{c}}}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}\;\\\quad\quad - \;{{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )y^{\prime}_{{\text{c}}} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )y^{\prime}_{{\text{c}}} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }} \hfill \\ \quad \quad + {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\sin \theta \gamma_{{\text{c}}} \lambda_{{\text{c}}} \phi_{1} (x_{1} )\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; \hfill \\ \end{gathered} $$
$$ k_{{417}} = - \lambda _{{\text{c}}} \int\limits_{0}^{1} {\varphi _{3} (x)\varphi _{3}^{\prime } (x)dx} \int\limits_{0}^{1} {\varphi _{3}^{{\prime 2}} (x)dx/2\beta _{{\text{c}}}^{2} h_{3} ;} $$
$$ k_{418} = - {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} (x_{1} )} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} (x_{1} )} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{419} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{3} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{3} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{420} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{2} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi^{\prime}_{2} (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{421} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {\varphi_{2}^{\prime 2} (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {\varphi_{2}^{\prime 2} (x)} dx} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{422} = - {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi_{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \cos \theta \int_{0}^{1} {\phi_{1} (x_{1} )\varphi_{1}^{\prime } (x)} dx\int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx} {\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{423} = - {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {\varphi_{1}^{\prime 2} (x)} dx} \mathord{\left/ {\vphantom {{\lambda_{{\text{c}}} \int_{0}^{1} {\varphi_{3} (x)\varphi_{3}^{\prime \prime } (x)} dx\int_{0}^{1} {\varphi_{1}^{\prime 2} (x)} dx} {2\beta_{{\text{c}}}^{2} h_{3} }}} \right. \kern-0pt} {2\beta_{{\text{c}}}^{2} h_{3} }}; $$
$$ k_{424} = {{k_{n} \varphi_{3} (l_{1} )} \mathord{\left/ {\vphantom {{k_{n} \varphi_{3} (l_{1} )} {h_{3} }}} \right. \kern-0pt} {h_{3} }}; $$
$$ k_{425} = {{c_{{\text{n}}} \varphi_{3} (l_{1} )} \mathord{\left/ {\vphantom {{c_{{\text{n}}} \varphi_{3} (l_{1} )} {h_{3} }}} \right. \kern-0pt} {h_{3} }}; $$
$$ k_{501} = {{k_{{\text{n}}} } \mathord{\left/ {\vphantom {{k_{{\text{n}}} } {m_{{\text{n}}} }}} \right. \kern-0pt} {m_{{\text{n}}} }}; $$
$$ k_{502} \; = {{c_{{\text{n}}} } \mathord{\left/ {\vphantom {{c_{{\text{n}}} } {m_{{\text{n}}} }}} \right. \kern-0pt} {m_{{\text{n}}} }}. $$

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Wang, Y., Kang, H., Cong, Y. et al. Nonlinear dynamic analysis of a cable-stayed beam with a nonlinear energy sink. Acta Mech 235, 1921–1944 (2024). https://doi.org/10.1007/s00707-023-03818-6

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