Abstract
A new formulation for the modular construction of flexible multibody systems is presented. By rearranging the equations for a flexible floating body and introducing the appropriate canonical momenta, the model is recast into a coupled system of ordinary and partial differential equations in port-Hamiltonian (pH) form. This approach relies on a floating frame description and is valid under the assumption of small deformations. This allows including mechanical models that cannot be easily formulated in terms of differential forms. Once a pH model is established, a finite element based method is then introduced to discretize the dynamics in a structure-preserving manner. Thanks to the features of the pH framework, complex multibody systems could be constructed in a modular way. Constraints are imposed at the velocity level, leading to an index 2 quasilinear differential-algebraic system. Numerical tests are carried out to assess the validity of the proposed approach.
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This work is supported by the project ANR-16-CE92-0028, entitled Interconnected Infinite-Dimensional systems for Heterogeneous Media, INFIDHEM, financed by the French National Research Agency (ANR) and the Deutsche Forschungsgemeinschaft (DFG). Further information is available at https://websites.isae-supaero.fr/infidhem/the-project.
Appendices
Appendix A: Mathematical tools
We recall here some identities and definitions that will be used throughout the paper.
1.1 A.1 Properties of the cross-product
We denote by \(\widetilde{\left [\boldsymbol{a}\right ]}\) the skew-symmetric map associated to vector \(\boldsymbol{a} = [a_{x}, a_{y}, a_{z}]^{\top }\), that is,
This map allows rewriting the cross-product as a matrix–vector product \(\boldsymbol{a}\wedge \boldsymbol{b} = \widetilde{\left [\boldsymbol{a}\right ]}\boldsymbol{b}\). The cross-product satisfies the anticommutativity property
Furthermore, it satisfies the Jacobi Identity
1.2 A.2 Adjoint of operators
In this paper, the adjoint of an operator is used. We recall the necessary definitions.
Definition 1
Given a linear operator \(\mathcal{A}: \mathscr{H}^{1} \rightarrow \mathscr{H}^{2}\) between Hilbert spaces, the adjoint \(\mathcal{A}^{*}:\mathscr{H}^{2}\rightarrow \mathscr{H}^{1}\) fulfills
To illustrate this definition, consider the operator \(\mathcal{I}^{\Omega }= \int _{\Omega }(\cdot ) \;\mathrm{d}\Omega : \mathscr{L}^{2}(\Omega , \mathbb{R}^{3}) \rightarrow \mathbb{R}^{3}\). Given a function \(\boldsymbol{u} \in \mathscr{L}^{2}(\Omega , \mathbb{R}^{3})\) and a vector \(\boldsymbol{v} \in \mathbb{R}^{3}\), the adjoint operator \((\mathcal{I}^{\Omega })^{*}\) extends the vector \(\boldsymbol{v}\) as a constant vector field over \(\Omega \),
Definition 2
A linear bounded operator \(\mathcal{A}: \mathscr{H} \rightarrow \mathscr{H}\) is self-adjoint if
Definition 3
A linear bounded operator \(\mathcal{A}: \mathscr{H} \rightarrow \mathscr{H}\) is skew-adjoint if
Indeed, the differential operators that appear in \(\boldsymbol{\mathcal{J}}\) (\(\operatorname{Div} , \operatorname{Grad} \)) are unbounded in the \(\mathscr{L}^{2}\) topology. Whenever unbounded operators are considered, it is important to define their domain. To avoid the need of specifying domains, the notion of formal (or essential) adjoint can be evoked. The formal adjoint respects the integration by parts formula and is defined only for sufficiently smooth functions with compact support. In this sense \(\operatorname{Div} , \operatorname{Grad} \) are formally skew-adjoint, since for smooth functions with compact support, it holds
The definition of the domain of the operators, which requires the knowledge of the boundary conditions, has not been specified. For this reason, the \(\boldsymbol{\mathcal{J}}\) operator is said to be formally skew-adjoint (or simply skew-symmetric).
1.3 A.3 Index of a differential-algebraic system
When dealing with differential-algebraic systems, an important notion is the index.
Definition 4
The index of a DAE is the minimum number of differentiation steps required to transform a DAE into an ODE.
Because of their structure, pH multibody systems are of index 2. Consider for simplicity a generic linear pH multibody system whose equations are
Matrix \(\mathbf{M}\) is squared and invertible and matrix \(\mathbf{G}\) is of full row rank. If the second equation is derived twice in time, then
Therefore, the system index is 2.
Appendix B: Detailed derivation of the equation of motions
The detailed derivation of the pH system (12) is presented here. We stick to the notation adopted along the paper. First, let us recall the equations for a floating flexible body reported in [44, 45].
-
Linear momentum balance:
$$ \begin{aligned} &m ^{i}\ddot{\boldsymbol{r}}_{P} + \boldsymbol{R} \widetilde{\left [\boldsymbol{s}_{u}\right ]}^{\top }\dot{\boldsymbol{\omega }}_{P} + \boldsymbol{R} \int _{\Omega } \rho \ddot{\boldsymbol{u}}_{f} \;\mathrm{d}{\Omega } = \\ &\quad + \boldsymbol{R} \left \{ - \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{s}_{u} - \int _{\Omega } 2 \rho \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \dot{\boldsymbol{u}}_{f} \;\mathrm{d}{\Omega } + \int _{\Omega } \boldsymbol{\beta }\;\mathrm{d}{\Omega } + \int _{\partial \Omega } \boldsymbol{\tau }\;\mathrm{d}{\Gamma } \right \} \end{aligned} $$(58) -
Angular momentum balance:
$$ \begin{aligned} \widetilde{\left [\boldsymbol{s}_{u}\right ]} { \boldsymbol{R}^{\top }} \ ^{i}\ddot{\boldsymbol{r}}_{P} + \boldsymbol{J}_{u} \dot{\boldsymbol{\omega }}_{P} + \int _{\Omega } \rho \widetilde{\left [\boldsymbol{x}_{f}\right ]} \ddot{\boldsymbol{u}}_{f} \;\mathrm{d}{\Omega } + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{J}_{u} \boldsymbol{\omega }_{P} = \\ - \int _{\Omega } 2\rho \widetilde{\left [\boldsymbol{x}_{f}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \dot{\boldsymbol{u}}_{f} \;\mathrm{d}{\Omega } + \int _{\Omega } \widetilde{\left [\boldsymbol{x}_{f}\right ]} \boldsymbol{\beta }\;\mathrm{d}{\Omega } + \int _{\partial \Omega } \widetilde{\left [\boldsymbol{x}_{f}\right ]} \boldsymbol{\tau }\;\mathrm{d}{\Gamma } \end{aligned} $$(59) -
Flexibility PDE:
$$ \rho {\boldsymbol{R}^{\top }} \ ^{i}\ddot{\boldsymbol{r}}_{P} + \rho ( \widetilde{\left [\dot{\boldsymbol{\omega }}_{P}\right ]} + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]})\boldsymbol{x}_{f} + \rho (2 \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \dot{\boldsymbol{u}}_{f} + \ddot{\boldsymbol{u}}_{f}) = \operatorname{Div}{\boldsymbol{\Sigma }} + \boldsymbol{\beta }, $$(60)
The first two equations are written in the inertial frame and so they need to be projected in the body frame. Considering that the position of point \(P\), i.e., \({}^{i}{\boldsymbol{r}}_{P}\), is computed in the inertial frame and \(\boldsymbol{v}_{P}\) in the body frame, one has \({}^{i}\dot{\boldsymbol{r}}_{P} = \boldsymbol{R} \boldsymbol{v}_{P}\). Taking the derivative of this gives
Substituting (61) into (58), (59), (60) and premultiplying (58) by \(\boldsymbol{R}^{\top }\), Eqs. (1) (2), and (3) are obtained.
-
Linear momentum balance:
$$ \begin{aligned} &m (\dot{\boldsymbol{v}}_{P} + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{v}_{P}) + \widetilde{\left [\boldsymbol{s}_{u}\right ]}^{\top }\dot{\boldsymbol{\omega }}_{P} + \int _{\Omega } \rho \dot{\boldsymbol{v}}_{f} \;\mathrm{d}{\Omega } = \\ &\quad - \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{s}_{u} - \int _{\Omega } 2 \rho \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} {\boldsymbol{v}}_{f} \;\mathrm{d}{\Omega } + \int _{\Omega } \boldsymbol{\beta }\;\mathrm{d}{\Omega } + \int _{ \partial \Omega } \boldsymbol{\tau }\;\mathrm{d}{\Gamma }. \end{aligned} $$(62) -
Angular momentum balance:
$$ \begin{aligned} \widetilde{\left [\boldsymbol{s}_{u}\right ]} ( \dot{\boldsymbol{v}}_{P} + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{v}_{P}) + \boldsymbol{J}_{u} \dot{\boldsymbol{\omega }}_{P} + \int _{\Omega } \rho \widetilde{\left [\boldsymbol{x}_{f}\right ]} \dot{\boldsymbol{v}}_{f} \;\mathrm{d}{\Omega } + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{J}_{u} \boldsymbol{\omega }_{P} = \\ - \int _{\Omega } 2\rho \widetilde{\left [\boldsymbol{x}_{f}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} {\boldsymbol{v}}_{f} \;\mathrm{d}{\Omega } + \int _{\Omega } \widetilde{\left [\boldsymbol{x}_{f}\right ]} \boldsymbol{\beta }\;\mathrm{d}{\Omega } + \int _{\partial \Omega } \widetilde{\left [\boldsymbol{x}_{f}\right ]} \boldsymbol{\tau }\;\mathrm{d}{\Gamma }. \end{aligned} $$(63) -
Flexibility PDE:
$$ \begin{aligned} \rho (\dot{\boldsymbol{v}}_{P} + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{v}_{P}) + \rho ( \widetilde{\left [\dot{\boldsymbol{\omega }}_{P}\right ]} + \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]})(\boldsymbol{x}_{f}) + \rho (2 \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} { \boldsymbol{v}}_{f} + \dot{\boldsymbol{v}}_{f}) = \\ \operatorname{Div}{\boldsymbol{\Sigma }} + \boldsymbol{\beta }, \end{aligned} $$(64)where \(\boldsymbol{v}_{f} = \dot{\boldsymbol{u}}_{f}\).
Consider now the term \(\widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{s}_{u}\), appearing in (62). Using the anticommutativity (53) and the fact that the cross-map is skew-symmetric, i.e., \(\widetilde{\left [\boldsymbol{a}\right ]} = - \widetilde{\left [\boldsymbol{a}\right ]}^{\top }\), one finds
Equation (62) is then rewritten as
The terms \(\widetilde{\left [\boldsymbol{s}_{u}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} \boldsymbol{v}_{P}, \; 2\rho \widetilde{\left [\boldsymbol{x}_{f}\right ]} \widetilde{\left [\boldsymbol{\omega }_{P}\right ]} {\boldsymbol{v}}_{f}\), appearing in (63), can be rewritten using the Jacobi identity (54) as
Equation (63) is then rewritten as
Notice that \(2 \widetilde{\left [\boldsymbol{v}_{f}\right ]}\boldsymbol{v}_{P} + 2 \widetilde{\left [\boldsymbol{v}_{P}\right ]}\boldsymbol{v}_{f} = 0\). Using again the anticommutativity, Eq. (64) is expressed as
Indeed, Eqs. (65), (68), (69) are exactly (5), (6), (7). Now by definitions (13), (14),
Equations (58), (59), (60) are written as
with
Hence, it is clear that Eqs. (58), (59), (60) from [44, 45] are equivalently recast in the form (12).
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Brugnoli, A., Alazard, D., Pommier-Budinger, V. et al. Port-Hamiltonian flexible multibody dynamics. Multibody Syst Dyn 51, 343–375 (2021). https://doi.org/10.1007/s11044-020-09758-6
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DOI: https://doi.org/10.1007/s11044-020-09758-6