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Internal resonance and energy transfer of a cable-stayed beam with a tuned mass damper

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Abstract

This study considers a novel nonlinear system, namely a cable-stayed beam with a tuned mass damper (cable-beam-TMD model), allowing the description of energy transfer among the beam, cable and TMD. In this system, the vibration of the TMD is involved and one-to-one-to-one internal resonance among the modes of the beam, cable and TMD is investigated when external primary resonance of the beam occurs. Galerkin’s method is utilized to discretize the equations of motion of the beam and cable. In this way, a set of ordinary differential equations are derived, which are solved by the method of multiple time scales. Then the steady-state solutions of the system are obtained by using Newton–Raphson method and continued by pseudo-arclength algorithm. The response curves, time histories and phase portraits are provided to explore the effect of the TMD on the nonlinear behaviors of the model. Meanwhile, a partially coupled system, namely a cable-beam-TMD model ignoring the vibration of the TMD, is also studied. The nonlinear characteristics of the two cases are compared with each other. The results reveal the occurrence of energy transfer among the beam, cable and TMD.

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Acknowledgements

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

Author information

Authors and Affiliations

  1. Key Laboratory of Disaster Prevention and Structural Safety of Ministry of Education, Guangxi University, Nanning, 530004, Guangxi, China

    Xiaoyang Su & Houjun Kang

  2. College of Civil Engineering, Hunan University, Changsha, 410082, Hunan, China

    Xiaoyang Su & Houjun Kang

  3. School of Civil Engineering and Architecture, Guangxi University, Nanning, 530004, Guangxi, China

    Xiaoyang Su, Houjun Kang, Tieding Guo & Yunyue Cong

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

    Xiaoyang Su, Houjun Kang, Tieding Guo & Yunyue Cong

  5. Guangxi Key Laboratory of Disaster Prevention and Engineering Safety, Guangxi University, Nanning, 530004, Guangxi, China

    Xiaoyang Su, Houjun Kang, Tieding Guo & Yunyue Cong

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  1. Xiaoyang Su
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Correspondence to Houjun Kang.

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Appendices

Appendix A

The expressions of the Galerkin integral coefficients in Eq. (20) are given as follows

$$ b_{11} = \omega_{b}^{2} = \frac{1}{{\int_{0}^{1} {\phi_{b}^{2} (s)} {\text{d}}s}}\left[ {\frac{{\int_{0}^{1} {\phi_{b} (s)\phi_{b}^{\prime \prime \prime \prime } (s)} {\text{d}}s}}{{\beta_{b}^{4} }} + \gamma_{c} \psi_{c} \phi_{b}^{2} (s_{1} )\sin \theta \cos \theta \int_{0}^{1} {y^{\prime }_{c} (x){\text{d}}x} + \gamma_{c} \psi_{c} \phi_{b}^{2} (s_{1} )\sin^{2} \theta } \right]; $$
$$ b_{12} = - \frac{{\gamma_{c}^{2} \psi_{c} \phi_{b}^{3} {(}s_{1} {\text{)cos}}^{2} \theta \sin \theta }}{{2\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}; $$
$$ b_{13} = - \frac{{\eta_{b} }}{{\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}\frac{{\int_{0}^{1} {\phi^{\prime}_{b} {(}s{)}\phi^{\prime}_{b} {(}s{)}} {\text{d}}s\int_{0}^{1} {\phi_{b} {(}s{)}\phi^{\prime\prime}_{b} {(}s{)}} {\text{d}}s}}{{2\beta_{b}^{4} }}; $$
$$ b_{14} = - \frac{{\psi_{c} \phi_{b} {(}s_{1} {\text{)sin}}\theta \int_{0}^{1} {y^{\prime}_{c} } {(}x{)}\phi^{\prime}_{c} {(}x{\text{)d}}x}}{{\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}; $$
$$ b_{15} = \frac{{\gamma_{c} \psi_{c} \phi_{b}^{2} {(}s_{1} {\text{)sin}}\theta \cos \theta \int_{0}^{1} {\phi^{\prime}_{c} {(}x{)}} {\text{d}}x}}{{\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}; $$
$$ b_{16} = - \frac{{\psi_{c} \phi_{b} {(}s_{1} {\text{)sin}}\theta \int_{0}^{1} {\phi^{\prime}_{c} {(}x{)}\phi^{\prime}_{c} {(}x{)}} {\text{d}}x}}{{2\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}; $$
$$ F = \int_{0}^{1} {F_{1} (s)\phi_{b} {(}s{)}} {\text{d}}s; $$

\(b_{17} = \frac{ - F}{{\int_{0}^{1} {\phi_{b}^{2} {(}s{)}} {\text{d}}s}}.\)

The expressions of the Galerkin integral coefficients in Eq. (21) are given as follows

$$ b_{21} = \omega_{c}^{2} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\left[ {K_{d} \phi_{c}^{2} {(}l_{1} {)} - \frac{{\lambda_{c} \int_{0}^{1} {y^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{\beta_{c}^{2} }} - \int_{0}^{1} {\frac{{\phi_{c} (x)\phi^{\prime\prime}_{c} (x)}}{{\beta_{c}^{2} }}} {\text{d}}x} \right]; $$
$$ b_{22} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\left[ {\mu_{c} \int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x + C_{d} \phi_{c}^{2} {(}l_{1} {)}} \right]; $$
$$ b_{23} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\left[ { - \gamma_{c} \mu_{c} \phi_{b} (s_{1} )\cos \theta \int_{0}^{1} {x\phi_{c} (x){\text{d}}x} - C_{d} l_{1} \gamma_{c} \phi_{b} (s_{1} )\phi_{c} (l_{1} )\cos \theta } \right] $$
$$ b_{24} = - \frac{{\gamma_{c} \phi_{b} (s_{1} )\cos \theta \int_{0}^{1} {x\phi_{c} (x){\text{d}}x} }}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ b_{25} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\left[ {\frac{{\lambda_{c} \gamma_{c} \phi_{b} (s_{1} )\cos \theta \int_{0}^{1} {y^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{\beta_{c}^{2} }} + \frac{{\lambda_{c} \gamma_{c} \phi_{b} (s_{1} )\sin \theta \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{\beta_{c}^{2} }} - K_{d} l_{1} \gamma_{c} \phi_{b} (s_{1} )\phi_{c} {(}l_{1} {\text{)cos}}\theta } \right]; $$
$$ b_{26} = - \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} (x)} {\text{d}}x}}\frac{{\lambda_{c} \gamma_{c}^{2} \phi_{b}^{2} (s_{1} )\cos^{2} \theta \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{2\beta_{c}^{2} }}; $$
$$ b_{{27}} = \frac{1}{{\int_{0}^{1} {\phi _{c}^{2} (x)} {\text{d}}x}}\left[ {\frac{{\lambda _{c} \gamma _{c} \phi _{b} (s_{1} )\cos \theta \int_{0}^{1} {\phi _{c}^{\prime } (x){\text{d}}x} \int_{0}^{1} {y_{c}^{{\prime \prime }} (x)\phi _{c} (x){\text{d}}x} }}{{\beta _{c}^{2} }} + \frac{{\lambda _{c} \gamma _{c} \phi _{b} (s_{1} )\cos \theta \int_{0}^{1} {y_{c}^{\prime } (x){\text{d}}x\int_{0}^{1} {\phi _{c} (x)\phi _{c}^{{\prime \prime }} (x){\text{d}}x} } }}{{\beta _{c}^{2} }}} \right.\left. {\quad + \frac{{\lambda _{c} \gamma _{c} \phi _{b} (s_{1} )\sin \theta \int_{0}^{1} {\phi _{c} (x)\phi _{c}^{{\prime \prime }} (x){\text{d}}x} }}{{\beta _{c}^{2} }}} \right]; $$
$$ b_{28} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\left[ { - \frac{{\lambda_{c} \int_{0}^{1} {\phi^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {y^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{2\beta_{c}^{2} }} - \frac{{\lambda_{c} \int_{0}^{1} {y^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x} \int_{0}^{1} {\phi^{\prime\prime}_{c} (x)\phi_{c} (x){\text{d}}x} }}{{\beta_{c}^{2} }}} \right]; $$
$$ b_{29} = - \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\frac{{\lambda_{c} \gamma_{c}^{2} \phi_{b}^{2} (s_{1} )\cos^{2} \theta \int_{0}^{1} {\phi_{c} (x)\phi^{\prime\prime}_{c} (x){\text{d}}x} }}{{2\beta_{c}^{2} }}; $$
$$ b_{210} = \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\frac{{\lambda_{c} \gamma_{c} \phi_{b} (s_{1} )\cos \theta \int_{0}^{1} {\phi^{\prime}_{c} (x){\text{d}}x\int_{0}^{1} {\phi_{c} (x)\phi^{\prime\prime}_{c} (x){\text{d}}x} } }}{{\beta_{c}^{2} }}; $$
$$ b_{211} = - \frac{1}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}\frac{{\lambda_{c} \int_{0}^{1} {\phi^{\prime}_{c} (x)\phi^{\prime}_{c} (x){\text{d}}x\int_{0}^{1} {\phi_{c} (x)\phi^{\prime\prime}_{c} (x){\text{d}}x} } }}{{2\beta_{c}^{2} }}; $$
$$ b_{212} = - \frac{{K_{d} \phi_{c} (l_{1} )}}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}; $$
$$ b_{213} = - \frac{{C_{d} \phi_{c} (l_{1} )}}{{\int_{0}^{1} {\phi_{c}^{2} {(}x{)}} {\text{d}}x}}; $$

The expressions of the Galerkin integral coefficients in Eq. (22) are given as follows

$$ b_{31} = 2\xi_{d} \omega_{d} ; $$
$$ b_{32} = l_{1} \gamma_{c} \omega_{d}^{2} \phi_{b} (s_{1} )\cos \theta ; $$
$$ b_{33} = - \omega_{d}^{2} \phi_{c} (l_{1} ); $$
$$ b_{34} = 2l_{1} \gamma_{c} \xi_{d} \omega_{d} \phi_{b} (s_{1} )\cos \theta ; $$
$$ b_{35} = - 2\xi_{d} \omega_{d} \phi_{c} (l_{1} ); $$

Appendix B

The expressions of the coefficients in Eq. (47) are defined as follows

$$ \Gamma_{{{11}}} { = }\frac{{b_{14} b_{25} }}{{4\omega_{b} \omega_{c} }}; $$
$$ \Gamma_{{{12}}} { = } - \frac{{b_{14} \sigma_{2} }}{{2\omega_{b} }}; $$
$$ \Gamma_{{{13}}} { = }\frac{{b_{14} b_{212} }}{{4\omega_{b} \omega_{c} }}; $$
$$ \Gamma_{{{14}}} { = } - 3b_{13} + \frac{{10b_{12}^{2} }}{{3\omega_{b}^{2} }} + \frac{{2b_{15} b_{26} }}{{\omega_{c}^{2} }} - \frac{{b_{15} b_{26} }}{{4\omega_{b}^{2} - \omega_{c}^{2} }}; $$
$$ \Gamma_{{{15}}} { = }\frac{{2b_{12} b_{15} }}{{\omega_{b}^{2} }} - \frac{{b_{15} b_{27} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{4b_{16} b_{26} }}{{\omega_{c}^{2} }} + \frac{{2b_{12} b_{15} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{12} b_{15} }}{{\omega_{c} (2\omega_{b} + \omega_{c} )}} - \frac{{b_{15} b_{27} }}{{\omega_{b} (\omega_{b} + 2\omega_{c} )}}; $$
$$ \Gamma_{{{16}}} { = } - \frac{{2b_{16} b_{27} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} - \frac{{b_{15} b_{28} }}{{3\omega_{c}^{2} }} + \frac{{b_{15}^{2} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{12} b_{16} }}{{ - \omega_{b}^{2} + 4\omega_{c}^{2} }}; $$
$$ \Gamma_{{{17}}} { = } - \frac{{b_{12} b_{15} }}{{3\omega_{b}^{2} }} - \frac{{b_{15} b_{27} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{2b_{12} b_{15} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{16} b_{26} }}{{4\omega_{b}^{2} - \omega_{c}^{2} }}; $$
$$ \Gamma_{{{18}}} { = }\frac{{4b_{12} b_{16} }}{{\omega_{b}^{2} }} - \frac{{2b_{16} b_{27} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{2b_{15} b_{28} }}{{\omega_{c}^{2} }} + \frac{{b_{15}^{2} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{b_{15}^{2} }}{{\omega_{c} (2\omega_{b} + \omega_{c} )}} - \frac{{2b_{16} b_{27} }}{{\omega_{b} (\omega_{b} + 2\omega_{c} )}}; $$
$$ \Gamma_{{{19}}} { = }\frac{{2b_{15} b_{16} }}{{\omega_{b}^{2} }} + \frac{{10b_{16} b_{28} }}{{3\omega_{c}^{2} }} - \frac{{b_{15} b_{16} }}{{ - \omega_{b}^{2} + 4\omega_{c}^{2} }}; $$

The expressions of the coefficients in Eq. (48) are defined as follows

$$ \Gamma_{{{21}}} { = }b_{24} \omega_{b}^{2} + \frac{{b_{25} \sigma_{2} }}{{2\omega_{c} }} + \frac{{b_{212} b_{32} }}{{4\omega_{c} \omega_{d} }}; $$
$$ \Gamma_{{{22}}} { = }\frac{{b_{14} b_{25} }}{{4\omega_{b} \omega_{c} }} + \frac{{b_{212} b_{33} }}{{4\omega_{c} \omega_{d} }}; $$
$$ \Gamma_{{{23}}} { = }\frac{{b_{212} \sigma_{3} }}{{2\omega_{c} }}; $$
$$ \Gamma_{{{24}}} { = }\frac{{10b_{12} b_{26} }}{{3\omega_{b}^{2} }} + \frac{{2b_{26} b_{27} }}{{\omega_{c}^{2} }} - \frac{{b_{26} b_{27} }}{{4\omega_{b}^{2} - \omega_{c}^{2} }}; $$
$$ \Gamma_{{{25}}} { = } - 2b_{29} + \frac{{2b_{12} b_{27} }}{{\omega_{b}^{2} }} - \frac{{b_{27}^{2} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{4b_{26} b_{28} }}{{\omega_{c}^{2} }} + \frac{{2b_{15} b_{26} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{15} b_{26} }}{{\omega_{c} (2\omega_{b} + \omega_{c} )}} - \frac{{b_{27}^{2} }}{{\omega_{b} (\omega_{b} + 2\omega_{c} )}}; $$
$$ \Gamma_{{{26}}} { = } - b_{210} - \frac{{2b_{27} b_{28} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} - \frac{{b_{27} b_{28} }}{{3\omega_{c}^{2} }} + \frac{{b_{15} b_{27} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{16} b_{26} }}{{ - \omega_{b}^{2} + 4\omega_{c}^{2} }}; $$
$$ \Gamma_{{{27}}} { = } - b_{29} - \frac{{b_{12} b_{27} }}{{3\omega_{b}^{2} }} - \frac{{b_{27}^{2} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{2b_{15} b_{26} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{2b_{26} b_{28} }}{{4\omega_{b}^{2} - \omega_{c}^{2} }}; $$
$$ \Gamma_{{{28}}} { = } - 2b_{210} + \frac{{4b_{16} b_{26} }}{{\omega_{b}^{2} }} - \frac{{2b_{27} b_{28} }}{{\omega_{b} (\omega_{b} - 2\omega_{c} )}} + \frac{{2b_{27} b_{28} }}{{\omega_{c}^{2} }} + \frac{{b_{15} b_{27} }}{{(2\omega_{b} - \omega_{c} )\omega_{c} }} - \frac{{b_{15} b_{27} }}{{\omega_{c} (2\omega_{b} + \omega_{c} )}} - \frac{{2b_{27} b_{28} }}{{\omega_{b} (\omega_{b} + 2\omega_{c} )}}; $$
$$ \Gamma_{{{29}}} { = } - 3b_{211} + \frac{{2b_{16} b_{27} }}{{\omega_{b}^{2} }} + \frac{{10b_{28}^{2} }}{{3\omega_{c}^{2} }} - \frac{{b_{16} b_{27} }}{{ - \omega_{b}^{2} + 4\omega_{c}^{2} }}; $$

The expressions of the coefficients in Eq. (49) are defined as follows

$$ \Gamma_{{{31}}} { = }\frac{{b_{32} \sigma_{2} }}{{2\omega_{d} }} - \frac{{b_{32} \sigma_{3} }}{{2\omega_{d} }} + \frac{{b_{25} b_{33} }}{{4\omega_{c} \omega_{d} }}; $$
$$ \Gamma_{{{32}}} { = } - \frac{{b_{33} \sigma_{3} }}{{2\omega_{d} }} + \frac{{b_{14} b_{32} }}{{4\omega_{b} \omega_{d} }}; $$
$$ \Gamma_{{{33}}} { = }\frac{{b_{212} b_{33} }}{{4\omega_{c} \omega_{d} }}. $$

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Su, X., Kang, H., Guo, T. et al. Internal resonance and energy transfer of a cable-stayed beam with a tuned mass damper. Nonlinear Dyn 110, 131–152 (2022). https://doi.org/10.1007/s11071-022-07644-8

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