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A three-sub-step composite method for the analysis of rigid body rotations with Euler parameters

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Abstract

This paper proposes a composite method for the analysis of rigid body rotations based on Euler parameters. The proposed method contains three sub-steps, wherein for keeping as much low-frequency information as possible the first two sub-steps adopt the trapezoidal rule, and the four-point backward interpolation formula is used in the last sub-step to flexibly control the amount of high-frequency dissipation. On this basis, in terms of the relation between Euler parameters and angular velocity, the stepping formulations of the proposed method are further modified for maximizing the accuracy of the angular velocity. For the analysis of rigid body rotations, the accuracy of the proposed method can converge to second-order, and the amount of its high-frequency dissipation can smoothly range from one (conservative scheme) to zero (annihilating scheme). Additionally, in the proposed method, the constraints at the displacement and velocity levels are strictly satisfied, and the numerical drifts at the acceleration level can be effectively eliminated. Furthermore, the proposed method is generalized to the field of rigid-flexible multibody systems described by Euler parameters, and in this work, its implementation procedure is provided. Several benchmark rigid body rotations and rigid-flexible multibody problems show the advantages of the proposed method in stability, accuracy, dissipation, efficiency, and energy conservation.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (12202058, 12172023, 11872090) the China Postdoctoral Science Foundation (2022M710386) and the Young Elite Scientists Sponsorship Program by BAST (BYESS2023344).

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  1. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Yi Ji

  2. Institute of Solid Mechanics, Beihang University, Beijing, 100083, China

    Yufeng Xing

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Appendix: Implementation of the mTTBIF in rigid-flexible multibody systems

Appendix: Implementation of the mTTBIF in rigid-flexible multibody systems

The generalized coordinates q of the rigid -flexible multibody systems governed by Eq. (70) can be defined as

$$ {\varvec{q}} = \left[ {\begin{array}{*{20}c} {{\varvec{x}}^{{\text{T}}} } & {{\boldsymbol{e}}^{{\text{T}}} } \\ \end{array} } \right]^{{\text{T}}} $$
(A.1)

where x and e respectively stand for position vector and Euler parameters. Since the vector x and the vector e can be decoupled, the motion equations given in Eq. (70) can be rewritten as

$$ \left[ {\begin{array}{*{20}c} {{\overline{\varvec{M}}}\left( {\varvec{x}} \right){\ddot{\varvec{x}}} - {\overline{\varvec{Q}}}\left( {{\dot{\varvec{x}}},{\varvec{x}},t} \right)} \\ {} \\ {} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {} \\ {{\tilde{\varvec{M}}}\left( {\boldsymbol{e}} \right){\ddot{\varvec{e}}} - {\tilde{\varvec{Q}}}\left( {{\dot{\boldsymbol{e}}},{\boldsymbol{e}},t} \right)} \\ {} \\ \end{array} } \right] + \left[ {\begin{array}{*{20}c} {{{\varvec{\varPhi}}}_{{\varvec{x}}}^{{\text{T}}} \left( {{\varvec{x}},{\boldsymbol{e}}} \right){{\varvec{\lambda}}}} \\ {{{\varvec{\varPhi}}}_{{\boldsymbol{e}}}^{{\text{T}}} \left( {{\varvec{x}},{\boldsymbol{e}}} \right){{\varvec{\lambda}}}} \\ {{{\varvec{\varPhi}}}\left( {{\varvec{x}},{\boldsymbol{e}}} \right)} \\ \end{array} } \right] = {\boldsymbol{0}} $$
(A.2)

When using the mTTBIF, the components of q with the Euler parameters are computed by

$$ \left\{ \begin{gathered} {\boldsymbol{e}}_{t + \gamma \Delta t} = {\boldsymbol{e}}_{t} + \gamma \Delta t{\dot{\boldsymbol{e}}}_{t} + 0.25\gamma^{2} \Delta t^{2} \left( {{\ddot{\varvec{e}}}_{t} + {\ddot{\varvec{e}}}_{t + \gamma \Delta t} } \right) \hfill \\ {\dot{\boldsymbol{e}}}_{t + \gamma \Delta t} = {\boldsymbol{L}}_{t + \gamma \Delta t}^{{\text{T}}} \left( {{\boldsymbol{L}}_{t} {\dot{\boldsymbol{e}}}_{t} + 0.5\gamma \Delta t\left( {{\boldsymbol{L}}_{t} {\ddot{\varvec{e}}}_{t} + {\boldsymbol{L}}_{t + \gamma \Delta t} {\ddot{\varvec{e}}}_{t + \gamma \Delta t} } \right)} \right) \hfill \\ \end{gathered} \right. $$
(A.3)
$$ \left\{ \begin{gathered} {\boldsymbol{e}}_{t + 2\gamma \Delta t} = {\boldsymbol{e}}_{t + \gamma \Delta t} + \gamma \Delta t{\dot{\boldsymbol{e}}}_{t + \gamma \Delta t} + 0.25\gamma^{2} \Delta t^{2} \left( {{\ddot{\varvec{e}}}_{t + \gamma \Delta t} + {\ddot{\varvec{e}}}_{t + 2\gamma \Delta t} } \right) \hfill \\ {\dot{\boldsymbol{e}}}_{t + 2\gamma \Delta t} = {\boldsymbol{L}}_{t + 2\gamma \Delta t}^{{\text{T}}} \left( {{\boldsymbol{L}}_{t + \gamma \Delta t} {\dot{\boldsymbol{e}}}_{t + \gamma \Delta t} + 0.5\gamma \Delta t\left( {{\boldsymbol{L}}_{t + \gamma \Delta t} {\ddot{\varvec{e}}}_{t + \gamma \Delta t} + {\boldsymbol{L}}_{t + 2\gamma \Delta t} {\ddot{\varvec{e}}}_{t + 2\gamma \Delta t} } \right)} \right) \hfill \\ \end{gathered} \right. $$
(A.4)
$$ \left\{ \begin{gathered} {\boldsymbol{e}}_{t + \Delta t} = {\boldsymbol{e}}_{t} + \Delta t\left[ {\left( {\theta_{0} + \theta_{3} } \right){\dot{\boldsymbol{e}}}_{t} + \theta_{1} {\dot{\boldsymbol{e}}}_{t + \gamma \Delta t} + \theta_{2} {\dot{\boldsymbol{e}}}_{t + 2\gamma \Delta t} + \theta_{3} \Delta t\left( {\theta_{0} {\ddot{\varvec{e}}}_{t} + \theta_{1} {\ddot{\varvec{e}}}_{t + \gamma \Delta t} + \theta_{2} {\ddot{\varvec{e}}}_{t + 2\gamma \Delta t} + \theta_{3} {\ddot{\varvec{e}}}_{t + \Delta t} } \right)} \right] \hfill \\ {\dot{\boldsymbol{e}}}_{t + \Delta t} = {\boldsymbol{L}}_{t + \Delta t}^{{\text{T}}} \left[ {{\boldsymbol{L}}_{t} {\dot{\boldsymbol{e}}}_{t} + \Delta t\left( {\theta_{0} {\boldsymbol{L}}_{t} {\ddot{\varvec{e}}}_{t} + \theta_{1} {\boldsymbol{L}}_{t + \gamma \Delta t} {\ddot{\varvec{e}}}_{t + \gamma \Delta t} + \theta_{2} {\boldsymbol{L}}_{t + 2\gamma \Delta t} {\ddot{\varvec{e}}}_{t + 2\gamma \Delta t} + \theta_{3} {\boldsymbol{L}}_{t + \Delta t} {\ddot{\varvec{e}}}_{t + \Delta t} } \right)} \right] \hfill \\ \end{gathered} \right. $$
(A.5)

and the remaining components of q are calculated by the classical TTBIF [41].

In the calculations, the increments of the three sub-steps can be obtained by the Newton–Raphson method in a uniform form, as follows:

$$ {\varvec{G}}_{\tau } \left( {\begin{array}{*{20}c} {\Delta {\ddot{\varvec{x}}}} \\ {\Delta {\ddot{\varvec{e}}}} \\ {\Delta \lambda } \\ \end{array} } \right) = \left( {\begin{array}{*{20}c} {{\overline{\varvec{M}}}\left( {\varvec{x}} \right){\ddot{\varvec{x}}} + {{\varvec{\varPhi}}}_{{\varvec{x}}}^{{\text{T}}} \left( {{\varvec{x}},{\boldsymbol{e}}} \right){{\varvec{\lambda}}} - {\overline{\varvec{Q}}}\left( {{\dot{\varvec{x}}},{\varvec{x}},\tau } \right)} \\ {{\tilde{\varvec{M}}}\left( {\boldsymbol{e}} \right){\ddot{\varvec{e}}} + {{\varvec{\varPhi}}}_{{\boldsymbol{e}}}^{{\text{T}}} \left( {{\varvec{x}},{\boldsymbol{e}}} \right){{\varvec{\lambda}}} - {\tilde{\varvec{Q}}}\left( {{\dot{\boldsymbol{e}}},{\boldsymbol{e}},\tau } \right)} \\ {{{\varvec{\varPhi}}}\left( {{\varvec{x}},{\boldsymbol{e}}} \right)} \\ \end{array} } \right)_{\tau } $$
(A.6)

where τ = t + γΔt, t + 2γΔt, or t + Δt, and the Jacobian matrix Gτ has the form as

$$ {\varvec{G}}_{\tau } = \left( {\begin{array}{*{20}c} {{\overline{\varvec{M}}} - \frac{{\partial {\overline{\varvec{Q}}}}}{{\partial {\dot{\varvec{x}}}}}\frac{{\partial {\dot{\varvec{x}}}}}{{\partial {\ddot{\varvec{x}}}}} + \left( {\frac{{\partial {\overline{\varvec{M}}}}}{{\partial {\varvec{x}}}}{\ddot{\varvec{x}}} + \frac{{\partial {{\varvec{\varPhi}}}_{{\varvec{x}}}^{{\text{T}}} }}{{\partial {\varvec{x}}}}{{\varvec{\lambda}}} - \frac{{\partial {\overline{\varvec{Q}}}}}{{\partial {\varvec{x}}}}} \right)\frac{{\partial {\varvec{x}}}}{{\partial {\ddot{\varvec{x}}}}}} & {} & {{{\varvec{\varPhi}}}_{{\varvec{x}}}^{{\text{T}}} } \\ {} & {{\tilde{\varvec{M}}} - \frac{{\partial {\tilde{\varvec{Q}}}}}{{\partial {\dot{\boldsymbol{e}}}}}\frac{{\partial {\dot{\boldsymbol{e}}}}}{{\partial {\ddot{\varvec{e}}}}} + \left( {\frac{{\partial {\tilde{\varvec{M}}}}}{{\partial {\boldsymbol{e}}}}{\ddot{\varvec{e}}} + \frac{{\partial {{\varvec{\varPhi}}}_{{\varvec{x}}}^{{\text{T}}} }}{{\partial {\boldsymbol{e}}}}{{\varvec{\lambda}}} - \frac{{\partial {\tilde{\varvec{Q}}}}}{{\partial {\boldsymbol{e}}}}} \right)\frac{{\partial {\boldsymbol{e}}}}{{\partial {\ddot{\varvec{e}}}}}} & {{{\varvec{\varPhi}}}_{{\boldsymbol{e}}}^{{\text{T}}} } \\ {{{\varvec{\varPhi}}}_{{\varvec{x}}}^{{}} \frac{{\partial {\varvec{x}}}}{{\partial {\ddot{\varvec{x}}}}}} & {{{\varvec{\Phi}}}_{{\boldsymbol{e}}}^{{}} \frac{{\partial {\boldsymbol{e}}}}{{\partial {\ddot{\varvec{e}}}}}} & {} \\ \end{array} } \right) $$
(A.7)

Furthermore, if the motions of flexible bodies are described by the singularity-free elements [50, 51], the generalized coordinate vector q = e only includes Euler parameters and the motion equations governed by Eq. (70) become

$$ \left\{ \begin{gathered} {\varvec{M}}\left( {\varvec{e}} \right){{\ddot{\varvec{e}}}} + {{\varvec{\varPhi}}}_{{\varvec{e}}}^{{\text{T}}} \left( {\varvec{e}} \right){{\varvec{\uplambda}}} = {\varvec{Q}}\left( {{\dot{\varvec{e}}},{\varvec{e}},t} \right) \hfill \\ {{\varvec{\varPhi}}}\left( {{\varvec{e}},t} \right) = {\varvec{0}} \hfill \\ \end{gathered} \right. $$
(A.8)

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Ji, Y., Xing, Y. A three-sub-step composite method for the analysis of rigid body rotations with Euler parameters. Nonlinear Dyn 111, 14309–14333 (2023). https://doi.org/10.1007/s11071-023-08410-0

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