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Vibration suppression of a cable-stayed beam by a nonlinear energy sink

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Abstract

The vibration control of cables of cable-stayed bridges has been one of the hot research fields in recent years. In this study, the influence of a nonlinear energy sink (NES) on vibration of the cable-stayed beam structure under external harmonic loads is investigated. Firstly, a mathematical model consisting of a cable, a beam, and a NES is established. By considering the interaction between the cable and beam, the nonlinear dynamic equation of the NES-cable-beam coupled system are derived by applying Hamilton's principle. Then, the motion equations of cable and beam are discretized into a set of ordinary differential equations (ODEs) using Galerkin method. The incremental harmonic balance (IHB) method is used to solve the ODEs. In addition, the results obtained by IHB method are verified those obtained using numerical method. Finally, the influence of NES on the response of the cable-stayed beam is investigated. Meantime, the influence of the cable-beam coupling vibration on the vibration mitigation effect of NES attached to the cable is analyzed. The results show that NES has a significant vibration mitigation effect on both cable and beam. However, the vibration mitigation effect of NES is reduced by the cable-beam coupling vibration. Furthermore, the stronger the degree of cable-beam coupling vibration, the worse the vibration mitigation effect of NES on the cable.

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Acknowledgements

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

Funding

The authors have not disclosed any funding.

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Authors and Affiliations

  1. State Key Laboratory of Featured Metal Materials and Life-cycle Safety for Composite Structures, 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. Research Center of Engineering Mechanics, Guangxi University, Nanning, 530004, China

    Yifei Wang, Houjun Kang, Yunyue Cong & Tieding Guo

  4. Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning, China

    Yifei Wang, Houjun Kang, Yunyue Cong & Tieding Guo

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

    Tao Fu

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  1. Yifei Wang
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  2. Houjun Kang
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Correspondence to Houjun Kang.

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Appendix 1

Appendix 1

The coefficients in Eqs. (22–26) are given as

$$ c_{101} = \mu_{b} ; $$
$$ c_{102} = \frac{{\int_{0}^{1} {\phi_{1} {(}x{)}\phi_{1}^{(4)} {(}x{)}} {\text{d}}x}}{{\beta_{b}^{4} \int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\sin \theta \phi_{1} {(}x_{1} {)}}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{c} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{103} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{c} {(}x{)}} \varphi^{\prime}_{1} {(}x{\text{)d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{104} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{c} {(}x{)}} \varphi^{\prime}_{2} {(}x{\text{)d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{105} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {y^{\prime}_{c} {(}x{)}} \varphi^{\prime}_{3} {(}x{\text{)d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{106} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}}}{{2\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{107} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{108} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{109} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{110} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{111} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{112} = - \frac{{\kappa_{c} \sin \theta \phi_{1} {(}x_{1} {)}\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{113} = - \frac{{\eta_{b} \int_{0}^{1} {\phi_{1}^{\prime 2} {(}x{)}} {\text{d}}x\int_{0}^{1} {\phi_{1} {(}x{)}\phi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{b}^{4} \int_{0}^{1} {\phi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{201} = \frac{{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{202} = \mu_{c} ; $$
$$ c_{203} = \frac{{\mu_{c} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{1} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{204} = - \frac{{\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x + \lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{205} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{206} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{207} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{208} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{209} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{210} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{211} = - \frac{{\lambda_{c} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{212} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{213} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{214} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y_{c}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{215} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{216} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{217} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{218} = - \frac{{\lambda_{c} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{219} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{220} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{221} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{222} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{223} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{1} {(}x{)}\varphi_{1}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{224} = \frac{{k_{n} \varphi_{1} (l_{1} )}}{{\int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{225} = \frac{{c_{n} \varphi_{1} (l_{1} )}}{{\int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{226} = \frac{{\int_{0}^{1} {f\varphi_{1} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{1}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{301} = \frac{{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{302} = \mu_{c} ; $$
$$ c_{303} = \frac{{\mu_{c} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{2} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{304} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{305} = - \frac{{\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x + \lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{306} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{307} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{308} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{309} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{310} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{311} = - \frac{{\lambda_{c} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{312} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{313} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{314} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{315} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y_{c}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{316} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{317} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{318} = - \frac{{\lambda_{c} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{319} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{320} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{321} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{322} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{323} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{2} {(}x{)}\varphi_{2}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{324} = \frac{{k_{n} \varphi_{2} (l_{1} )}}{{\int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{325} = \frac{{c_{n} \varphi_{2} (l_{1} )}}{{\int_{0}^{1} {\varphi_{2}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{401} = \frac{{\cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{402} = \mu_{c} ; $$
$$ c_{403} = \frac{{\mu_{c} \cos \theta \phi_{1} {(}x_{1} {)}\int_{0}^{1} {x\varphi_{3} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{404} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{405} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{406} = - \frac{{\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x + \lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{407} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\phi_{1} {(}x_{1} {)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{408} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{409} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{410} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{411} = - \frac{{\lambda_{c} \cos^{2} \theta \phi_{1}^{2} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{412} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{413} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {y_{c}^{\prime } {(}x{)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{414} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{415} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{416} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}y_{c}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}} - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}y_{c}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}} + \frac{{\sin \theta \gamma_{c} \lambda_{c} \phi_{1} {(}x_{1} {)}\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{417} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{418} = - \frac{{\lambda_{c} \cos^{2} \theta \int_{0}^{1} {\phi_{1}^{2} {(}x_{1} {)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{419} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{3}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{420} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{2}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{421} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{2}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{422} = - \frac{{\lambda_{c} \cos \theta \int_{0}^{1} {\phi_{1} {(}x_{1} {)}\varphi_{1}^{\prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x}}{{\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{423} = - \frac{{\lambda_{c} \int_{0}^{1} {\varphi_{3} {(}x{)}\varphi_{3}^{\prime \prime } {(}x{)}} {\text{d}}x\int_{0}^{1} {\varphi_{1}^{\prime 2} {(}x{)}} {\text{d}}x}}{{2\beta_{c}^{2} \int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{424} = \frac{{k_{n} \varphi_{3} (l_{1} )}}{{\int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{425} = \frac{{c_{n} \varphi_{3} (l_{1} )}}{{\int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{426} = \frac{{\int_{0}^{1} {f\varphi_{3} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\varphi_{3}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ c_{501} = \frac{{k_{n} }}{{m_{n} }}; $$
$$ c_{502} = \frac{{c_{n} }}{{m_{n} }}. $$

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Wang, Y., Kang, H., Cong, Y. et al. Vibration suppression of a cable-stayed beam by a nonlinear energy sink. Nonlinear Dyn 111, 14829–14849 (2023). https://doi.org/10.1007/s11071-023-08651-z

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