Modeling and analysis of a flexible spinning Euler-Bernoulli beam with centrifugal stiffening and softening: A Linear Fractional Representation approach with application to spinning spacecraft

TL;DR

The paper develops a six-DOF analytical, boundary-condition-agnostic model for a spinning, flexible Euler-Bernoulli beam using Linear Fractional Representation within the TITOP framework. The model captures bending, traction, torsion, and centrifugal effects, and is parameterized as an LPV system driven by angular velocity to reflect centrifugal stiffening. Validation against literature and NASTRAN demonstrates accurate natural frequencies and dynamics, with convergence as model granularity increases. A THOR/Cluster-inspired spinning spacecraft case study shows the approach scales to complex multibody systems and supports robust control design for spinning, flexible structures.

Abstract

The derivation of a linear fractional representation (LFR) model for a flexible, spinning and uniform Euler-Bernoulli beam is accomplished using the {Lagrange} technique, fully capturing the centrifugal force generated by the spinning motion and accounting for its dependence on the angular velocity. This six degrees of freedom (DOF) model accounts for the behavior of deflection in the moving body frame, encompassing the bending, traction and torsion dynamics. The model is also designed to be compliant with the Two-Input-Two-Output Port (TITOP) approach, which offers the possibility to model complex multibody mechanical systems, while keeping the uncertain nature of the plant and condensing all the possible mechanical configurations in a single LFR. To evaluate the effectiveness of the model, various scenarios are considered and their results are tabulated. These scenarios include uniform beams with fixed root boundary conditions for different values of tip mass, root offset and angular velocity. The results from the analysis of the uniform cantilever beam are compared with solutions found in the literature and obtained from a commercial finite element software. Ultimately, this paper presents a multibody model for a spinning spacecraft mission scenario. A comprehensive analysis of the system dynamics is conducted, providing insights into the behavior of the spacecraft under spinning conditions.

Figures (11)

• Figure 1: A spinning satellite with various rigid and/or flexible appendages.
• Figure 2: (a) Three-Dimensional (3D) representation of a flexible spinning beam $\mathcal{A}_i$. (b) TITOP model ${\left[\mathfrak{T}_{P_i C_i}^{\mathcal{A}_i}(\mathrm{s})\right]}_{\mathcal{R}_{a_i}}$ block-diagram.
• Figure 3: Block-diagram representation of $[\mathfrak{R}_{PC}^{\mathcal{A}}]_{\mathcal{R}_a}$: the linear $24\times 24$ model of a rigid body $\mathcal{A}$ computed at the points $P$ and $C$.
• Figure 4: Detailed block-diagram representation of ${\left[\mathfrak{X}_{B}^{\mathcal{B}}(\mathrm s)\right]_{\mathcal{R}_{b}}}$.
• Figure 5: (a) Parameterization of the beam deflection at a given time $t$ (Note: the displacement of a specific point $M(x)$ on the deformed flexible beam ${u}(x,t)$ along the $\mathbf{x}_a$-axis is displayed for both cases): (a) bending in the plane $\left(P, \mathbf{x}_a, \mathbf{y}_a\right)$. (b) bending in the plane $\left(P, \mathbf{x}_a, \mathbf{z}_a\right)$.