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Nonlinear free vibrations of Timoshenko–Ehrenfest beams using finite element analysis and direct scheme

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Abstract

In this work, nonlinear free vibrations of fully geometrically exact Timoshenko–Ehrenfest beams are investigated. First, the exact strong form of the Timonshenko–Ehrenfest beam, considering the geometrical nonlinearity, is derived, and the required formulations are obtained. Since the strong forms of governing equations are highly nonlinear, a nonlinear finite element analysis (FEA) is employed to obtain the weak form. The FEA is utilized to compute natural frequencies and mode shapes; the direct scheme is adopted to solve the eigenvalue problem which is obtained by eliminating nonlinear terms. Then, each eigenvector is normalized, and the nonlinear stiffness matrix is derived and the nonlinear free vibration analysis is carried out. A recursive procedure is adopted to proceed until the convergence criterion is satisfied. Finally, the applicability of the proposed formulation is provided with some examples and results are compared with those available in the literature.

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Data can be made available upon reasonable request.

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Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Institute of Structural Mechanics, Bauhaus-University Weimar, Weimar, Germany

    Nasser Firouzi & Timon Rabczuk

  2. Department of Civil and Building Engineering, and Architecture (DICEA), Polytechnic University of Marche, Via Brecce Bianche, 60131, Ancona, Italy

    Stefano Lenci

  3. School of Engineering, Westlake University, Hangzhou, People’s Republic of China

    Marco Amabili

  4. Department of Mechanical Engineering, McGill University, Montreal, PQ, H3A 2K6, Canada

    Marco Amabili

Authors
  1. Nasser Firouzi
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  2. Stefano Lenci
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  3. Marco Amabili
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  4. Timon Rabczuk
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Contributions

Conceptualization was done by NF and SL, formulation derivation and analyses were performed by NF. Data preparation was done by SL, MA, and TR. The first draft of the manuscript was written by NF. Supervision and editing the manuscript were done by MA, TR, and SL. All authors commented and reviewed the manuscript and approved the final version of it.

Corresponding author

Correspondence to Nasser Firouzi.

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Appendix

Appendix

It is known that in FE derivation, it is convenient to work in natural coordinates. The relation between two different coordinates is as follows:

$$ {\text{d}}x = \frac{{{\text{d}}x}}{{{\text{d}}\xi }}{\text{d}}\xi = J{\text{d}}\xi ,\,\,\,\,\,\,\,\,\,\,\frac{{{\text{d}}\left( * \right)}}{{{\text{d}}x}} = \frac{{{\text{d}}\left( * \right)}}{{{\text{d}}\xi }}\frac{{{\text{d}}\xi }}{{{\text{d}}x}} = J^{ - 1} \frac{{{\text{d}}\left( * \right)}}{{{\text{d}}\xi }} $$
(36)

The arrays of the stiffness and mass matrices are derived as follows:

$$ \left( {K_{uu} } \right)_{11} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(37)
$$ \left( {K_{uu} } \right)_{12} = \left( {K_{uu} } \right)_{21} = - \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(38)
$$ \left( {K_{uu} } \right)_{22} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{1}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{1}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(39)
$$ \left( {K_{uw} } \right)_{11} = - \frac{{K_{S} \left( {\theta_{1}^{e} + \theta_{2}^{e} + 2\arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{h_{e} - u_{1}^{e} + u_{2}^{e} }}} \right)} \right)}}{{2\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(40)
$$ \left( {K_{uw} } \right)_{12} = \left( {K_{uw} } \right)_{21} = \frac{{K_{S} \left( {\theta_{1}^{e} + \theta_{2}^{e} + 2\arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{h_{e} - u_{1}^{e} + u_{2}^{e} }}} \right)} \right)}}{{2\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(41)
$$ \left( {K_{uw} } \right)_{22} = - \frac{{K_{S} \left( {\theta_{1}^{e} + \theta_{2}^{e} + 2\arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{h_{e} - u_{1}^{e} + u_{2}^{e} }}} \right)} \right)}}{{2\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(42)
$$ \left( {K_{u} } \right)_{11} = - \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{{}} }} - \frac{1}{2} $$
(43)
$$ \left( {K_{u} } \right)_{12} = - \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{{}} }} - \frac{1}{2} $$
(44)
$$ \left( {K_{u} } \right)_{21} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{{}} }} - \frac{1}{2} $$
(45)
$$ \left( {K_{u} } \right)_{22} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{{}} }} - \frac{1}{2} $$
(46)
$$ \left( {K_{ww} } \right)_{11} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(47)
$$ \left( {K_{ww} } \right)_{12} = \left( {K_{ww} } \right)_{21} = - \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(48)
$$ \left( {K_{ww} } \right)_{22} = \frac{{K_{0} }}{{h_{e}^{{}} \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{{2h_{e}^{2} }} - \frac{1}{{2h_{e}^{{}} }} $$
(49)
$$ \left( {K_{w\theta } } \right)_{11} = \frac{{K_{S} h_{e} \left( {u_{1}^{e} - u_{2}^{e} - h_{e} } \right)}}{{2\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{2} $$
(50)
$$ \left( {K_{w\theta } } \right)_{12} = \frac{{K_{S} h_{e} \left( {u_{1}^{e} - u_{2}^{e} - h_{e} } \right)}}{{2\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{2} $$
(51)
$$ \left( {K_{w\theta } } \right)_{21} = - \frac{{K_{S} h_{e} \left( {u_{1}^{e} - u_{2}^{e} - h_{e} } \right)}}{{2\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{2} $$
(52)
$$ \left( {K_{w\theta } } \right)_{22} = - \frac{{K_{S} h_{e} \left( {u_{1}^{e} - u_{2}^{e} - h_{e} } \right)}}{{2\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }} - \left( {u_{1}^{e} - u_{2}^{e} } \right) + \frac{{\sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } }}{2} $$
(53)
$$ \left( {K_{w} } \right)_{11} = - \frac{{K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{\left( {h_{e} - u_{1}^{e} + u_{2}^{e} } \right)^{2} }}} \right)}}{{\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(54)
$$ \left( {K_{w} } \right)_{12} = - \frac{{K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{\left( {h_{e} - u_{1}^{e} + u_{2}^{e} } \right)^{2} }}} \right)}}{{\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(55)
$$ \left( {K_{w} } \right)_{21} = \frac{{K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{\left( {h_{e} - u_{1}^{e} + u_{2}^{e} } \right)^{2} }}} \right)}}{{\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(56)
$$ \left( {K_{w} } \right)_{22} = \frac{{K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{\left( {h_{e} - u_{1}^{e} + u_{2}^{e} } \right)^{2} }}} \right)}}{{\sqrt {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} } }} $$
(57)
$$ \left( {K_{\theta \theta } } \right)_{11} = - \frac{{K_{S} \left( {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \,\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \,\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + 3\frac{{K_{2} }}{{K_{S} }}} \right)}}{{3\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right)}} $$
(58)
$$ \left( {K_{\theta \theta } } \right)_{12} = \left( {K_{\theta \theta } } \right)_{21} = \frac{{ - K_{S} \left( {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \,\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \,\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} - 6\frac{{K_{2} }}{{K_{S} }}} \right)}}{{6\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right)}} $$
(59)
$$ \left( {K_{\theta \theta } } \right)_{22} = - \frac{{K_{S} \left( {h_{e}^{2} - 2h_{e} \left( {u_{1}^{e} - u_{2}^{e} } \right) + \,\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} + \,\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + 3\frac{{K_{2} }}{{K_{S} }}} \right)}}{{3\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right)}} $$
(60)
$$ \left( {K_{\theta } } \right)_{11} = \frac{1}{2}K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{u_{1}^{e} - u_{2}^{e} - h_{e} }}} \right)\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e}^{{}} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right) $$
(61)
$$ \left( {K_{\theta } } \right)_{12} = \left( {K_{\theta } } \right)_{21} = \frac{1}{2}K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{u_{1}^{e} - u_{2}^{e} - h_{e} }}} \right)\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e}^{{}} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right) $$
(62)
$$ \left( {K_{\theta } } \right)_{12} = \frac{1}{2}K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{u_{1}^{e} - u_{2}^{e} - h_{e} }}} \right)\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e}^{{}} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right) $$
(63)
$$ \left( {K_{\theta } } \right)_{22} = \frac{1}{2}K_{S} \arctan \left( {\frac{{w_{1}^{e} - w_{2}^{e} }}{{u_{1}^{e} - u_{2}^{e} - h_{e} }}} \right)\left( {\frac{{\left( {u_{1}^{e} - u_{2}^{e} } \right)^{2} }}{{h_{e}^{{}} }} - 2\left( {u_{1}^{e} - u_{2}^{e} } \right) + \sqrt {\left( {w_{1}^{e} - w_{2}^{e} } \right)^{2} + h_{e}^{2} } } \right) $$
(64)
$$ \begin{gathered} \left( {K_{W\theta } } \right)_{11} = \left( {K_{W\theta } } \right)_{12} = \left( {K_{W\theta } } \right)_{21} = \left( {K_{W\theta } } \right)_{22} = 0 \hfill \\ \left( {K_{UW} } \right)_{11} = \left( {K_{UW} } \right)_{12} = \left( {K_{UW} } \right)_{21} = \left( {K_{UW} } \right)_{22} = 0 \hfill \\ \left( {K_{\theta W} } \right)_{11} = \left( {K_{\theta W} } \right)_{12} = \left( {K_{\theta W} } \right)_{21} = \left( {K_{\theta W} } \right)_{22} = 0 \hfill \\ \left( {K_{\theta U} } \right)_{11} = \left( {K_{\theta U} } \right)_{12} = \left( {K_{\theta U} } \right)_{21} = \left( {K_{\theta U} } \right)_{22} = 0 \hfill \\ \end{gathered} $$
(65)
$$ \left( {M_{uu} } \right)_{11} = \left( {M_{uu} } \right)_{22} = \frac{{h_{e} m_{0} }}{3},\,\,\,\left( {M_{uu} } \right)_{12} = \left( {M_{uu} } \right)_{21} = \frac{{h_{e} m_{0} }}{6} $$
(66)
$$ \left( {M_{ww} } \right)_{11} = \left( {M_{ww} } \right)_{22} = \frac{{h_{e} m_{0} }}{3},\,\,\,\left( {M_{ww} } \right)_{12} = \left( {M_{ww} } \right)_{21} = \frac{{h_{e} m_{0} }}{6} $$
(67)
$$ \left( {M_{\theta \theta } } \right)_{11} = \left( {M_{\theta \theta } } \right)_{22} = \frac{{h_{e} m_{2} }}{3},\,\,\,\,\left( {M_{\theta \theta } } \right)_{12} = \left( {M_{\theta \theta } } \right)_{21} = \frac{{h_{e} m_{2} }}{6} $$
(68)

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Firouzi, N., Lenci, S., Amabili, M. et al. Nonlinear free vibrations of Timoshenko–Ehrenfest beams using finite element analysis and direct scheme. Nonlinear Dyn 112, 7199–7213 (2024). https://doi.org/10.1007/s11071-024-09403-3

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