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Comparison of finite element methods for dynamic analysis about rotating flexible beam

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Abstract

Dynamic analysis of flexible body under large rotation becomes increasingly important in many engineering applications. This work takes rotating flexible beam as numerical example under assumptions of straight beam, small deformation and fixed axis rotation. Results of six finite element methods are compared: floating frame of reference formulation (FFRF), generalized component mode synthesis (GCMS), total Lagrangian formulation (TLF), linear Euler–Bernoulli beam (LEBB), nonlinear Euler–Bernoulli beam (NEBB) and absolute nodal coordinate formulation (ANCF). Due to the effect of geometric nonlinearity or centrifugal stiffening, the results of linear methods FFRF, GCMS and LEBB are different to geometrically nonlinear methods TLF, NEBB and ANCF. Three points of view are obtained through numerical result analysis. Firstly, based on the equivalence between FFRF and GCMS, a special type of parametrically excited nonlinear system is equivalent to linear system, so as to analyze the Lyapunov stability of its solution. Secondly, for the ANCF, the system invariant matrix is presented to calculate elastic force, which avoids element integration or assembling in each time step, and then the computational efficiency is improved by at least an order of magnitude when compared to adopting element invariant matrix. Thirdly, for GCMS, TLF and ANCF, the velocity of relative coordinate for deformation is adopted to calculate linear internal damping force, which gets different deflection result in comparison with adopting velocity of absolute coordinate for deformation, under large rotational speed.

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The presented methods are implemented in C++, and the results of numerical examples can be replicated on the basis of supplementary material.

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Acknowledgements

The authors gratefully acknowledge the open-source library Eigen (http://eigen.tuxfamily.org/index.php?title=Main_Page) from eigenproject team and the open source library Spectra (http://spectralib.org/index.html) from Yixuan Qiu.

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Authors and Affiliations

  1. College of Mechanical Engineering, Shenyang Ligong University, Shenyang, 110159, China

    Quancheng Peng & Minghao Li

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  1. Quancheng Peng
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  2. Minghao Li
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All authors contributed to the study conception and design. The first draft of the manuscript was written by Quancheng Peng, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Quancheng Peng.

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Supplementary Information

Appendix

Appendix

$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rr}}e}} = \left[ {\begin{array}{cc} {{M_{\mathrm{{rr}}1e}}}&{}0\\ 0&{}{{M_{\mathrm{{rr}}1e}}} \end{array}} \right] + \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}2e}}}\\ {{\varvec{M}_{\mathrm{{rr}}3e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}4e}}}&{{\varvec{M}_{\mathrm{{rr}}5e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{W}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{W}_{\mathrm{{f}}e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}6e}}}&{}{{\varvec{M}_{\mathrm{{rr}}7e}}}\\ {{\varvec{M}_{\mathrm{{rr}}8e}}}&{}{{\varvec{M}_{\mathrm{{rr}}9e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{W}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{W}_{\mathrm{{f}}e}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(70)
$$\begin{aligned}{} & {} \begin{aligned} {M_{\mathrm{{rr}}1e}} = {l_e} \cdot \rho \cdot \int _0^1 {{X_0^2 \cdot A + {I_Z}}\mathrm{{d}}\xi } \end{aligned} \end{aligned}$$
(71)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rr}}2e}} = \\ {}&{l_e} \cdot \rho \cdot \int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} {{X_0} \cdot A}&{}{ - {S_Z}}\\ {{S_Z}}&{}{{X_0} \cdot A}\\ 0&{}0\\ { - {I_{YZ}}}&{}{ - {X_0} \cdot {S_Y}}\\ {{X_0} \cdot {S_Y}}&{}{ - {I_{YZ}}}\\ { - {X_0} \cdot {S_Z}}&{}{{I_Z}} \end{matrix}}} \right] \mathrm{{d}}\xi } \end{aligned} \end{aligned}$$
(72)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rr}}3e}} = \\ {}&{l_e} \cdot \rho \cdot \int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} {{S_Z}}&{}{{X_0} \cdot A}\\ { - {X_0} \cdot A}&{}{{S_Z}}\\ 0&{}0\\ {{X_0} \cdot {S_Y}}&{}{ - {I_{YZ}}}\\ {{I_{YZ}}}&{}{{X_0} \cdot {S_Y}}\\ { - {I_Z}}&{}{ - {X_0 \cdot S_Z}} \end{matrix}}} \right] \mathrm{{d}}\xi } \end{aligned} \end{aligned}$$
(73)
$$\begin{aligned}{} & {} \begin{aligned} {\varvec{M}_{\mathrm{{rr}}4e}} = {\varvec{M}_{\mathrm{{rr}}2e}^\mathrm{{T}}} \end{aligned} \end{aligned}$$
(74)
$$\begin{aligned}{} & {} \begin{aligned} {\varvec{M}_{\mathrm{{rr}}5e}} = {\varvec{M}_{\mathrm{{rr}}3e}^\mathrm{{T}}} \end{aligned} \end{aligned}$$
(75)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rr}}6e}} = {l_e}\cdot \rho \cdot \\ {}&\int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} A&{}0&{}0&{}0&{}{{S_Y}}&{}{ - {S_Z}}\\ 0&{}A&{}0&{}{ - {S_Y}}&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0\\ 0&{}{ - {S_Y}}&{}0&{}{{I_Y}}&{}0&{}0\\ {{S_Y}}&{}0&{}0&{}0&{}{{I_Y}}&{}{ - {I_{YZ}}}\\ { - {S_Z}}&{}0&{}0&{}0&{}{ - {I_{YZ}}}&{}{{I_Z}} \end{matrix}}} \right] \cdot {\varvec{N}_\mathrm{{E}}}\mathrm{{(}}\xi \mathrm{{)d}}\xi } \end{aligned} \end{aligned}$$
(76)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rr}}7e}} = {l_e}\cdot \rho \cdot \\ {}&\int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} 0&{}{ - A}&{}0&{}{{S_Y}}&{}0&{}0\\ A&{}0&{}0&{}0&{}{{S_Y}}&{}{ - {S_Z}}\\ 0&{}0&{}0&{}0&{}0&{}0\\ { - {S_Y}}&{}0&{}0&{}0&{}{ - {I_Y}}&{}{{I_{YZ}}}\\ 0&{}{ - {S_Y}}&{}0&{}{{I_Y}}&{}0&{}0\\ 0&{}{{S_Z}}&{}0&{}{ - {I_{YZ}}}&{}0&{}0 \end{matrix}}} \right] \cdot {\varvec{N}_\mathrm{{E}}}\mathrm{{(}}\xi \mathrm{{)d}}\xi } \end{aligned} \end{aligned}$$
(77)
$$\begin{aligned}{} & {} \begin{aligned} {\varvec{M}_{\mathrm{{rr}}8e}} = -{\varvec{M}_{\mathrm{{rr}}7e}} \end{aligned} \end{aligned}$$
(78)
$$\begin{aligned}{} & {} \begin{aligned} {\varvec{M}_{\mathrm{{rr}}9e}} = {\varvec{M}_{\mathrm{{rr}}6e}} \end{aligned} \end{aligned}$$
(79)
$$\begin{aligned}{} & {} \begin{aligned} {\varvec{M}_{\mathrm{{rf}}e}} = \tilde{\varvec{A}} \cdot {\varvec{M}_{\mathrm{{rf}}1e}} + \tilde{\varvec{A}} \cdot \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \cdot \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rf}}2e}}}\\ {{\varvec{M}_{\mathrm{{rf}}3e}}} \end{array}} \right] \end{aligned} \nonumber \\ \end{aligned}$$
(80)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rf1}}e}} = {l_e}\cdot \rho \cdot \\ {}&\int _0^1 {\left[ {{\begin{matrix} {{X_0}A}&{}{{S_Z}}&{}0&{}{ - {I_{YZ}}}&{}{{X_0}{S_Y}}&{}{ - {X_0}{S_Z}}\\ { - {S_Z}}&{}{{X_0}A}&{}0&{}{ - {X_0}{S_Y}}&{}{ - {I_{YZ}}}&{}{{I_Z}} \end{matrix}}} \right] \cdot {N_\mathrm{{E}}}\mathrm{{(}}\xi \mathrm{{)d}}\xi } \end{aligned} \end{aligned}$$
(81)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rf}}2e}} = {l_e}\cdot \rho \cdot \\ {}&\int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} A&{}0&{}0&{}0&{}{{S_Y}}&{}{ - {S_Z}}\\ 0&{}A&{}0&{}{ - {S_Y}}&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0\\ 0&{}{ - {S_Y}}&{}0&{}{{I_Y}}&{}0&{}0\\ {{S_Y}}&{}0&{}0&{}0&{}{{I_Y}}&{}{ - {I_{YZ}}}\\ { - {S_Z}}&{}0&{}0&{}0&{}{ - {I_{YZ}}}&{}{{I_Z}} \end{matrix}}} \right] \cdot {\varvec{N}_\mathrm{{E}}}\mathrm{{(}}\xi \mathrm{{)d}}\xi } \end{aligned} \end{aligned}$$
(82)
$$\begin{aligned}{} & {} \begin{aligned}&{\varvec{M}_{\mathrm{{rf}}3e}} = {l_e}\cdot \rho \cdot \\ {}&\int _0^1 {\varvec{N}_\mathrm{{E}}^\mathrm{{T}}\mathrm{{(}}\xi \mathrm{{)}} \cdot \left[ {{\begin{matrix} 0&{}A&{}0&{}{ - {S_Y}}&{}0&{}0\\ { - A}&{}0&{}0&{}0&{}{ - {S_Y}}&{}{{S_Z}}\\ 0&{}0&{}0&{}0&{}0&{}0\\ {{S_Y}}&{}0&{}0&{}0&{}{{I_Y}}&{}{ - {I_{YZ}}}\\ 0&{}{{S_Y}}&{}0&{}{ - {I_Y}}&{}0&{}0\\ 0&{}{ - {S_Z}}&{}0&{}{{I_{YZ}}}&{}0&{}0 \end{matrix}}} \right] \cdot {\varvec{N}_\mathrm{{E}}}\mathrm{{(}}\xi \mathrm{{)d}}\xi } \end{aligned} \end{aligned}$$
(83)
$$\begin{aligned}{} & {} \begin{aligned}&{\dot{\varvec{M}}_{\mathrm{{rr}}e}} = \left[ {\begin{array}{cc} {\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}2e}}}\\ {{\varvec{M}_{\mathrm{{rr}}3e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}4e}}}&{{\varvec{M}_{\mathrm{{rr}}5e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\dot{\varvec{W}}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\dot{\varvec{W}}_{\mathrm{{f}}e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}6e}}}&{}{{\varvec{M}_{\mathrm{{rr}}7e}}}\\ {{\varvec{M}_{\mathrm{{rr}}8e}}}&{}{{\varvec{M}_{\mathrm{{rr}}9e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{W}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{W}_{\mathrm{{f}}e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}6e}}}&{}{{\varvec{M}_{\mathrm{{rr}}7e}}}\\ {{\varvec{M}_{\mathrm{{rr}}8e}}}&{}{{\varvec{M}_{\mathrm{{rr}}9e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\dot{\varvec{W}}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\dot{\varvec{W}}_{\mathrm{{f}}e}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(84)
$$\begin{aligned}{} & {} \begin{aligned}&{\dot{\varvec{M}}_{\mathrm{{rf}}e}} = \dot{\tilde{\varvec{A}}} \cdot {\varvec{M}_{\mathrm{{rf}}1e}} + \dot{\tilde{\varvec{A}}} \cdot \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \cdot \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rf}}2e}}}\\ {{\varvec{M}_{\mathrm{{rf}}3e}}} \end{array}} \right] \\ {}&+ \tilde{\varvec{A}} \cdot \left[ {\begin{array}{cc} {\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\dot{\varvec{W}}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \cdot \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rf}}2e}}}\\ {{\varvec{M}_{\mathrm{{rf}}3e}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(85)
$$\begin{aligned}{} & {} \begin{aligned} \frac{{\partial {T_e}}}{{\partial {q_{ei}}}} = \frac{1}{2} \cdot \dot{\varvec{q}}_e^\mathrm{{T}} \cdot \frac{{\partial {\varvec{M}_e}}}{{\partial {q_{ei}}}} \cdot {\dot{\varvec{q}}_e} \end{aligned} \end{aligned}$$
(86)

where \(q_{ei}\) is the i-th component of vector \(\varvec{q}_e\).

$$\begin{aligned} \begin{aligned}&\frac{{\partial {\varvec{M}_{\mathrm{{rr}}e}}}}{{\partial {W_{\mathrm{{f}}ei}}}} = \left[ {\begin{array}{cc} {\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{c} {{\varvec{M}_{\mathrm{{rr}}2e}}}\\ {{\varvec{M}_{\mathrm{{rr}}3e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}4e}}}&{{\varvec{M}_{\mathrm{{rr}}5e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{H}_{\mathrm{{f}}ei}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{H}_{\mathrm{{f}}ei}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}6e}}}&{}{{\varvec{M}_{\mathrm{{rr}}7e}}}\\ {{\varvec{M}_{\mathrm{{rr}}8e}}}&{}{{\varvec{M}_{\mathrm{{rr}}9e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{W}_{\mathrm{{f}}e}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{W}_{\mathrm{{f}}e}}} \end{array}} \right] \\&\quad + \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{M}_{\mathrm{{rr}}6e}}}&{}{{\varvec{M}_{\mathrm{{rr}}7e}}}\\ {{\varvec{M}_{\mathrm{{rr}}8e}}}&{}{{\varvec{M}_{\mathrm{{rr}}9e}}} \end{array}} \right] \left[ {\begin{array}{cc} {{\varvec{H}_{\mathrm{{f}}ei}}}&{}\varvec{0}\\ \varvec{0}&{}{{\varvec{H}_{\mathrm{{f}}ei}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(87)

where \(W_{\mathrm{{f}}ei}\) is the i-th component of vector \(\varvec{W}_{\mathrm{{f}}e}\), \({\varvec{H}_{\mathrm{{f}}ei}} = \frac{{\partial {\varvec{W}_{\mathrm{{f}}e}}}}{{\partial {W_{\mathrm{{f}}ei}}}}\).

$$\begin{aligned} \begin{aligned} \frac{{\partial {\varvec{M}_{\mathrm{{rf}}e}}}}{{\partial {q_{\mathrm{{r}}i}}}} = \frac{{\partial \tilde{\varvec{A}}}}{{\partial {q_{\mathrm{{r}}i}}}} \cdot {\varvec{M}_{\mathrm{{rf1}}e}} + \frac{{\partial \tilde{\varvec{A}}}}{{\partial {q_{\mathrm{{r}}i}}}} \cdot \left[ {\begin{array}{cc} {\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{W}_{\mathrm{{f}}e}^\mathrm{{T}}} \end{array}} \right] \cdot \left[ {\begin{array}{c} {{\varvec{M}_{\mathrm{{rf2}}e}}}\\ {{\varvec{M}_{\mathrm{{rf3}}e}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(88)

where \(q_{\mathrm{{r}}i}\) is the i-th component of vector \(\varvec{q}_\mathrm{{r}}\).

$$\begin{aligned} \begin{aligned} \frac{{\partial {\varvec{M}_{\mathrm{{rf}}e}}}}{{\partial {W_{\mathrm{{f}}ei}}}} = \tilde{\varvec{A}} \cdot \left[ {\begin{array}{cc} {\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}}&{}\varvec{0}\\ \varvec{0}&{}{\varvec{H}_{\mathrm{{f}}ei}^\mathrm{{T}}} \end{array}} \right] \cdot \left[ {\begin{array}{c} {{\varvec{M}_{\mathrm{{rf2}}e}}}\\ {{\varvec{M}_{\mathrm{{rf3}}e}}} \end{array}} \right] \end{aligned} \end{aligned}$$
(89)

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Peng, Q., Li, M. Comparison of finite element methods for dynamic analysis about rotating flexible beam. Nonlinear Dyn 111, 13753–13779 (2023). https://doi.org/10.1007/s11071-023-08568-7

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