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Vibration behavior analysis of novelty corrugated-core sandwich plate structure by using first-order shear deformation plate and shell theories

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Abstract

A unified dynamic model is established for investigating the vibration behaviors of novelty corrugated-core sandwich plate structure (NCSPS) under various boundary conditions based on the first-order shear deformation plate and shell theories by selecting differential quadrature finite element method (DQFEM) in this article. The NCSPS is composed of two different elements including plate element and circular arc shell element and the coupling between the above elements is realized by coordinate transformation and common differential quadrature nodes. The penalty function method is selected to simulate the boundary conditions of NCSPS and five boundary conditions including free, simply supported (I), clamped (I), simply supported (II) and clamped (II) are taken into account in this paper. The convergence of the established model is investigated from the perspectives of differential quadrature nodes and penalty factors. The validations including accuracy, stability and universality of the established model are studied by comparing the results calculated by the established model with the corresponding results of ABAQUS. The free and forced vibration characteristics with regard to NCSPS subject to various boundary conditions are investigated detailly in terms of structure parameters. The investigations of free vibration behaviors of NCSPS under free and clamped (I) boundary conditions are realized by investigating the effect of structure parameters of NCSPS on its the natural frequency. For forced vibration analysis, only steady-state response is considered and the investigations of forced vibration behaviors of NCSPS under clamped (I) and (II) boundary conditions are performed by analyzing the influence of structure parameters of NCSPS on its displacement response.

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This manuscript has associated data in a data repository. [Authors’ comment: All data included in this manuscript are available upon request by contacting with the corresponding author.]

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Acknowledgements

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 51705537), State Key Laboratory of High Performance Complex Manufacturing, Central South University, China (Grant No. ZZYJKT2021-07). The authors also gratefully acknowledge the supports from Key Laboratory of Vibration and Control of Aero-Propulsion System, Ministry of Education, Northeastern University (rant No. VCAME202006).

Author information

Authors and Affiliations

  1. State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha, 410083, People’s Republic of China

    Zhen Li, Qingshan Wang, Rui Zhong & Wen Shao

  2. Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic & Transportation Engineering, Central South University, Changsha, 410075, People’s Republic of China

    Bin Qin

  3. Joint International Research Laboratory of Key Technology for Rail Traffic Safety, Central South University, Changsha, 410075, People’s Republic of China

    Bin Qin

  4. National & Local Joint Engineering Research Center of Safety Technology for Rail Vehicle, Central South University, Changsha, 410075, People’s Republic of China

    Bin Qin

Authors
  1. Zhen Li
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  2. Bin Qin
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  3. Qingshan Wang
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  4. Rui Zhong
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  5. Wen Shao
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Corresponding author

Correspondence to Qingshan Wang.

Appendix A

Appendix A

This appendix contains the stiffness matrices of plate and circular arc shell elements with arbitrary conditions.

$$ {\mathbf{K}}_{p11}^{i} = {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A1)
$$ {\mathbf{K}}_{p12}^{i} = \mu_{p} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A2)
$$ {\mathbf{K}}_{p21}^{i} { = }\mu_{p} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A3)
$$ {\mathbf{K}}_{p22}^{i} { = }{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A4)
$$ {\mathbf{K}}_{p33}^{i} { = }\frac{{h_{pi}^{2} }}{12}{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}\kappa_{p} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{h_{pi}^{2} }}{12}\frac{{{1} - \mu_{p} }}{2}{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A5)
$$ {\mathbf{K}}_{p34}^{i} { = }\frac{{h_{pi}^{2} }}{12}\left( {\mu_{p} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} } \right) $$
(A6)
$$ {\mathbf{K}}_{p35}^{i} { = }\frac{{{1} - \mu_{p} }}{2}\kappa_{p} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A7)
$$ {\mathbf{K}}_{p43}^{i} = \frac{{h_{pi}^{2} }}{12}\left( {\mu_{p} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} { + }\frac{{{1} - \mu_{p} }}{2}{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} } \right) $$
(A8)
$$ {\mathbf{K}}_{p44}^{i} = \frac{{h_{pi}^{2} }}{12}{\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} + \frac{{{1} - \mu_{p} }}{2}\kappa_{p} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)}} + \frac{{h_{pi}^{2} }}{12}\frac{{{1} - \mu_{p} }}{2}{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A9)
$$ {\mathbf{K}}_{p45}^{i} = \frac{{{1} - \mu_{p} }}{2}\kappa_{p} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A10)
$$ {\mathbf{K}}_{p53}^{i} = - \frac{{1 - \mu_{p} }}{2}\kappa_{p} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A11)
$$ {\mathbf{K}}_{p54}^{i} = - \frac{{1 - \mu_{p} }}{2}\kappa_{p} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{E}}_{{p{2}}}^{{\left( {1} \right)}} $$
(A12)
$$ {\mathbf{K}}_{p55}^{i} = \frac{{1 - \mu_{p} }}{2}\kappa_{p} \left( {{\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{A}}_{{p{2}}}^{{\left( {1} \right)}} + {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)T}} {\mathbf{C}}_{{p{2}}} {\mathbf{B}}_{{p{2}}}^{{\left( {1} \right)}} } \right) $$
(A13)
$$ {\mathbf{K}}_{s11}^{i} { = }{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{1 - \mu_{s} }}{{2R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A14)
$$ {\mathbf{K}}_{s12}^{i} { = }\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} + \frac{{1 - \mu_{s} }}{{2R_{si} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} $$
(A15)
$$ {\mathbf{K}}_{s13}^{i} { = }\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A16)
$$ {\mathbf{K}}_{s14}^{i} { = }{\mathbf{K}}_{s15}^{i} { = }{\mathbf{0}} $$
(A17)
$$ {\mathbf{K}}_{s21}^{i} { = }\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{1 - \mu_{s} }}{{2R_{si} }}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A18)
$$ {\mathbf{K}}_{s22}^{i} { = }\frac{1}{{R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} + \frac{{1 - \mu_{s} }}{2}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si}^{2} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A19)
$$ {\mathbf{K}}_{s23}^{i} { = }\frac{1}{{R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{E}} - \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si}^{2} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A20)
$$ {\mathbf{K}}_{s24}^{i} { = }{\mathbf{0}} $$
(A21)
$$ {\mathbf{K}}_{s25}^{i} { = } - \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A22)
$$ {\mathbf{K}}_{s31}^{i} { = }\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} $$
(A23)
$$ {\mathbf{K}}_{s32}^{i} { = }\frac{1}{{R_{si}^{2} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} - \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A24)
$$ {\mathbf{K}}_{s33}^{i} { = }\frac{1}{{R_{si}^{2} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} + \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{2}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A25)
$$ {\mathbf{K}}_{s34}^{i} { = }\frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{2}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A26)
$$ {\mathbf{K}}_{s35}^{e} { = }\frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A27)
$$ {\mathbf{K}}_{s41}^{i} { = }{\mathbf{K}}_{s42}^{i} { = }{\mathbf{0}} $$
(A28)
$$ {\mathbf{K}}_{s43}^{i} { = }\frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{2}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} $$
(A29)
$$ {\mathbf{K}}_{s44}^{i} { = }\frac{{h_{si}^{2} }}{12}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{h_{si}^{2} }}{12}\frac{{\left( {1 - \mu_{s} } \right)}}{{2R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} + \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{2}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A30)
$$ {\mathbf{K}}_{s45}^{i} { = }\frac{{h_{si}^{2} }}{12}\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} + \frac{{h_{si}^{2} }}{12}\frac{{\left( {1 - \mu_{s} } \right)}}{{2R_{si} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} $$
(A31)
$$ {\mathbf{K}}_{s51}^{i} { = }{\mathbf{0}} $$
(A32)
$$ {\mathbf{K}}_{s52}^{i} { = } - \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A33)
$$ {\mathbf{K}}_{s53}^{i} { = }\frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{{2R_{si} }}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A34)
$$ {\mathbf{K}}_{s54}^{i} { = }\frac{{h_{si}^{2} }}{12}\frac{{\mu_{s} }}{{R_{si} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{h_{si}^{2} }}{12}\frac{{\left( {1 - \mu_{s} } \right)}}{{2R_{si} }}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} $$
(A35)
$$ {\mathbf{K}}_{s55}^{i} { = }\frac{{h_{si}^{2} }}{12}\frac{1}{{R_{si}^{2} }}{\mathbf{B}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{B}}_{s2}^{\left( 1 \right)} + \frac{{h_{si}^{2} }}{12}\frac{{\left( {1 - \mu_{s} } \right)}}{2}{\mathbf{A}}_{s2}^{\left( 1 \right)T} {\mathbf{C}}_{s2} {\mathbf{A}}_{s2}^{\left( 1 \right)} + \frac{{\left( {1 - \mu_{s} } \right)\kappa_{s} }}{2}{\mathbf{E}}^{T} {\mathbf{C}}_{s2} {\mathbf{E}} $$
(A36)

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Li, Z., Qin, B., Wang, Q. et al. Vibration behavior analysis of novelty corrugated-core sandwich plate structure by using first-order shear deformation plate and shell theories. Eur. Phys. J. Plus 137, 270 (2022). https://doi.org/10.1140/epjp/s13360-022-02478-0

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  • DOI: https://doi.org/10.1140/epjp/s13360-022-02478-0

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