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Aeroelastic Analysis Considering the Coupling Effect Between the Reference and Elastic Displacements in Flexible Multibody Dynamics

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Abstract

Multibody system simulation has been widely used to model and analyze complicated dynamical systems. Many recent multibody system packages can simulate rigid and flexible bodies effectively and reliably, and offer a variety of alternative formulations. Despite numerous advances in modeling of flexible system, the well-known floating frame of reference formulation (FFRF) remains the most often used formulation. Although FFRF normally only permits modest elastic deformations to be included, this assumption is sufficient for many practical applications. Furthermore, when implemented with proper model order reduction techniques and adequate management of system inertia terms through the use of so-called inertia form integrals, and integration of external forces, FFRF is computationally efficient. In this study, the governing equations are derived from the integration of floating frame method in multibody dynamics and component mode synthesis in structural dynamics. The time-dependent aerodynamic force is taken into the governing equations. The equation manifests the time-dependent coupling between the reference and elastic displacement as well as between structure and aerodynamics force. Deriving the system of equations of motion for FFRF bodies is, in fact, a difficult and error-prone operation. The main goal of this work is to provide a reliable and universal set of inertia terms and aerodynamic force as external force terms within the FFRF. Additionally, a worked-out example has also been given to clarify the effects of the components on the response of flexible system.

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Data availability

The data generated during and/or analyzed during the current study are available within the manuscript and its supplementary file.

Abbreviations

\({\mathbf{r}}_{p}\) :

The position vector of an arbitrary point P on the body i in the multibody system, m

\({\mathbf{R}}\) :

The global position vector of the origin of the body reference, m

\({\mathbf{u}}\) :

The local position of point in the global coordinate

\({\mathbf{A}}\) :

The transformation matrix from the body coordinate to the global coordinate

\({\overline{\mathbf{u}}}\) :

The position vector of point in the body coordinate

\({\overline{\mathbf{u}}}_{0}\) :

The undeformed position vector in the body coordinate

\({\overline{\mathbf{u}}}_{f}\) :

The elastic deformation vector in the body coordinate

\({\mathbf{S}}\) :

Shape function matrix

\({\mathbf{Q}}_{v}\) :

Quadratic velocity vector, N

\({\mathbf{Q}}_{g}\) :

Gravitational force vector

\({\mathbf{Q}}_{e}\) :

External force vector

\({\mathbf{Q}}_{c}\) :

Constraint force vector

\({\mathbf{v}}_{b}\) :

Velocity in body coordinate

\({\alpha }_{r}\) :

Rigid-body angle of attack

\({\alpha }_{xe}\) :

Torsion of section

ρ :

Air Density, kg/m3

\(U\) :

Air speed, m/s

\({\mathbf{B}}_{l,m}\) :

Aerodynamic damping matrices

\({\mathbf{C}}_{l,m}\) :

Aerodynamic stiffness matrices

\(\delta {\mathbf{r}}_{c}\) :

Virtual displacement in global coordinate

\(\delta {{\varvec{\upkappa}}}\) :

Virtual orientation in global coordinate

\(P\) :

Point P

\(r\) :

Reference components

\(f\) :

Flexible components

\(I\) , \(b\) , \(e\) :

Global, body (reference), element coordinate

\(i\) :

Body number in the multibody system

\(j\) :

Element \(j\) on a deformable body \(i\)

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Acknowledgements

This work was supported by the BK 21 FOUR program through the National Research Foundation of Korea (NRF) funded by the Korean government (Grant No. 519990714521) and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant No. 2022R1A6A1A03056784).

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Authors and Affiliations

  1. Department of Smart Air Mobility, Korea Aerospace University, Goyang, South Korea

    Phuc Anh Nguyen

  2. Mechanical and Aerospace Department, Korea Aerospace University, Goyang, South Korea

    Hong-Gi Rho & Bong-Do Pyeon

  3. School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang-si, Gyeonggi-do, 10540, South Korea

    Moon-Sung Cho & Jae-Sung Bae

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Correspondence to Jae-Sung Bae.

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Nguyen, P.A., Rho, HG., Pyeon, BD. et al. Aeroelastic Analysis Considering the Coupling Effect Between the Reference and Elastic Displacements in Flexible Multibody Dynamics. Int. J. Aeronaut. Space Sci. 25, 468–486 (2024). https://doi.org/10.1007/s42405-023-00660-x

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  • DOI: https://doi.org/10.1007/s42405-023-00660-x

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