# Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes

## Abstract

This paper investigates the dynamics of a real-life aerospace structure possessing a strongly nonlinear component with multiple mechanical stops. A full-scale finite element model is built for gaining additional insight into the nonlinear dynamics that was observed experimentally, but also for uncovering additional nonlinear phenomena, such as quasiperiodic regimes of motion. Forced/unforced, damped/undamped numerical simulations are carried out using advanced techniques and theoretical concepts such as numerical continuation and nonlinear normal modes.

### Keywords

Aerospace structure Piecewise-linear nonlinearities Numerical continuation Nonlinear normal modes Modal interactions## 1 Introduction

It is widely accepted that virtually all engineering structures are nonlinear, at least in certain regimes of motion. Even if the common industrial practice is to ignore nonlinearity, a recent trend is to exploit them for engineering design, e.g., for vibration absorption and mitigation [1, 2, 3]. The last decade witnessed progresses in this direction, and, in particular, in the analysis of nonlinear aerospace structures. Experimental identification of nonlinearity during aircraft and helicopter ground vibration tests was, for instance, performed in references [4, 5, 6, 7, 8]. Nonlinearity was also evidenced and identified during spacecraft testing [9, 10]. However, most of the existing experimental contributions assumed or observed weakly nonlinear behaviors. In parallel, substantial efforts were made to address the numerical modeling of complex, nonlinear aerospace structures (see, e.g., [11, 12]). Analysis using advanced numerical continuation techniques was also carried out in [13, 14].

Very few studies attempted to numerically analyze and experimentally compare the dynamics of a real-life structure in strongly nonlinear regimes of motion. This is the main contribution of the present paper. The identification of the SmallSat spacecraft, a satellite possessing a nonlinear component with multiple axial and lateral mechanical stops, was achieved in [15] using measurements collected during a typical qualification test campaign. This study revealed that the spacecraft may exhibit complex dynamical phenomena in commonly endured experimental conditions. For instance, jumps, interactions between modes with noncommensurate linear frequencies, force relaxation, and chattering during impacts on the mechanical stops were reported in [15]. Furthermore, several interactions between local and global modes of the structure evidenced energy transfers to the payload, which jeopardize its structural integrity and, in turn, the satellite’s mission. Understanding and predicting these phenomena are thus of the utmost importance.

This paper builds a full-scale computational model of the satellite for gaining further insight into the observed nonlinear dynamics, but also for uncovering additional nonlinear phenomena not reproduced experimentally. Forced/unforced, damped/undamped numerical simulations are carried out using advanced techniques and theoretical concepts such as numerical continuation [16, 17] and nonlinear normal modes [18, 19]. We note that a formal model updating process could not be achieved during the test campaign. Bringing the predictions of the model in very close quantitative agreement with the experimental results is therefore not the objective of this paper.

The paper is organized as follows. A detailed finite element model of the underlying linear satellite is first built in Sect. 2 and reduced using the Craig–Bampton technique. The model identified experimentally for the nonlinear vibration isolation device is presented and incorporated in the finite element model in Sect. 3. The nonsmooth nonlinearities in the model are regularized for facilitating the ensuing numerical simulations. Sect. 4 provides the numerical evidence of some of the phenomena observed experimentally. A bifurcation analysis then reveals the existence of quasiperiodic regimes of motion. Section 5 carries out nonlinear modal analysis of the SmallSat spacecraft. It discusses in great detail the behavior of several nonlinear modes exhibiting nonlinear modal interactions and energy localization. The conclusions of this study are drawn in Sect. 6.

## 2 The SmallSat spacecraft structure

The spacecraft structure supports a dummy telescope mounted on a baseplate through a tripod; its mass is around 140 kg. The dummy telescope plate is connected to the SmallSat top floor by three shock attenuators, termed shock attenuation systems for spacecraft and adaptor (SASSAs) [21], whose dynamic behavior may exhibit nonlinearity. Besides, as depicted in Fig. 1b, a support bracket connects to one of the eight walls the so-called wheel elastomer mounting system (WEMS), which is loaded with an 8-kg dummy inertia wheel. The WEMS acts as a mechanical filter, which mitigates high-frequency disturbances coming from the inertia wheel through the presence of a soft elastomeric interface between its mobile part, i.e., the inertia wheel and a supporting metallic cross, and its fixed part, i.e the bracket and by extension the spacecraft. Moreover, the WEMS incorporates eight mechanical stops, covered with a thin layer of elastomer and designed to limit the axial and lateral motions of the inertia wheel during launch, which gives rise to strongly nonlinear dynamical phenomena (cf. Sect. 3).

### 2.1 Finite element modeling of the underlying linear satellite

A finite element model (FEM) of the SmallSat satellite created in the LMS-SAMTECH SAMCEF software is used in the present study to conduct numerical experiments. The model is presented in Fig. 1c, and it comprises about 150,000 degrees of freedom (DOFs). It idealizes the composite tube structure using orthotropic shell elements. The top floor, the bracket, and the wheel support are also modeled using shell elements. Boundary conditions are enforced at the base of the satellite through four clamped nodes. Proportional damping using the parameters provided by EADS-Astrium is also introduced in the model.

Comparison between numerical and experimental natural frequencies

Mode # | Model freq. (Hz) | Experimental freq. (Hz) |
---|---|---|

1 | 8.06 | 8.19 |

2 | 9.14 | – |

3 | 20.44 | – |

4 | 21.59 | – |

5 | 22.05 | 20.18 |

6 | 28.75 | 22.45 |

7 | 32.49 | – |

8 | 34.78 | 34.30 |

9 | 39.07 | – |

10 | 40.78 | 43.16 |

11 | 45.78 | 45.99 |

12 | 57.76 | 55.71 |

13 | 68.99 | 64.60 |

14 | 75.14 | – |

15 | 79.82 | – |

16 | 83.36 | – |

17 | 89.01 | 88.24 |

18 | 95.30 | – |

### 2.2 Reduced-order modeling

Because the WEMS nonlinearities are spatially localized, condensation of the linear FEM can be effectively achieved using the Craig–Bampton reduction technique [22]. This leads to a substantial decrease in the computational burden without degrading the accuracy of the numerical simulations in the frequency range of interest. The Craig–Bampton method expresses the complete set of initial DOFs in terms of retained DOFs and internal vibration modes of the structure clamped on the retained nodes. To introduce the WEMS nonlinearities, the reduced-order model (ROM) is constructed by keeping one node on both sides of the lateral and axial mechanical stops. In total, eight nodes of the initial FEM possessing 3 DOFs each and 10 internal modes of vibration are kept; this reduced model possesses 34 DOFs and is termed ROM810. For local excitation of the WEMS, a second ROM, termed ROM910, is created with an additional node on the metallic cross.

## 3 Modeling of the WEMS nonlinearities

Figure 4a presents a simplified, yet relevant, modeling of the WEMS where the inertia wheel, owing to its important rigidity, is seen as a point mass. The four nonlinear connections (NCs) between the WEMS mobile and fixed parts are labeled NC1-4, respectively.

Parameters of the WEMS nonlinear connections (adimensionalised for confidentiality)

Stiffness | NC1 | NC2 | NC4 | NC3 |
---|---|---|---|---|

Axial \(k_{Z}\) | 8.30 | 9.21 | 9.18 | 10.03 |

Lateral \(k_{X}\) | 1.31 | 1.31 | 0.69 | 0.69 |

Lateral \(k_{Y}\) | 0.69 | 0.69 | 0.69 | 0.69 |

Axial \(k_{+,Z}\) | 79.40 | 88.41 | 79.40 | 88.41 |

Axial \(k_{-,Z}\) | 118.07 | 116.73 | 118.07 | 116.73 |

Lateral \(k_{\pm ,XY}\) | 40 | 40 | 40 | 40 |

| ||||

Axial \(a_{+,Z}\) | 1.55 | 1.62 | 1.59 | 1.59 |

Axial \(a_{-,Z}\) | 1.01 | 0.84 | 0.93 | 0.93 |

Lateral \(a_{\pm ,XY}\) | 2 | 2 | 2 | 2 |

## 4 Direct numerical integration and numerical continuation

## 5 Nonlinear modal analysis of the SmallSat spacecraft

In the previous section, a nonconservative FEM was utilized to further investigate the nonlinear phenomena observed during the testing campaign of the SmallSat spacecraft. Because the damped dynamics can also be interpreted based on the topological structure and the bifurcations of the nonlinear normal modes (NNMs) of the underlying conservative system [28], a detailed nonlinear modal analysis is carried out herein.

An extension of Rosenberg’s definition is considered, i.e., an NNM is defined as a (nonnecessarily synchronous) periodic motion of the unforced, conservative system. The algorithm proposed in [29], which combines shooting and pseudoarclength continuation, is applied to the ROM810 model for NNM computation. Due to the frequency-energy dependence of nonlinear oscillations, NNMs are depicted in a frequency-energy plot (FEP). An NNM is represented by a point in the FEP, drawn at a frequency corresponding to the minimal period of the periodic motion, and at an energy equal to the conserved total energy during the motion. A branch depicted by a solid line represents the complete frequency-energy dependence of the considered mode.

^{1}The reason for this complex topology is that the dynamics has to evolve from NNM6, a mode with a predominant axial motion between the WEMS and the bracket activating a unique axial nonlinear connection, NC2-Z, to NNM17, a mode with lateral motion of the bracket activating two other nonlinear connections in the lateral direction, NC3-Y and NC4-Y. To understand this progression, the motion of the center of gravity of the WEMS cross is displayed in Fig. 15. Clearly, the WEMS motion takes place in the XZ plane at points A and B, YZ plane at point E, Y direction at points F, YZ plane at point H and finally back to XZ plane at point J. In addition, Table 3 displays the nonlinear connections that are active at the considered points together with the penetration in the corresponding regularization intervals.

Activation of the nonlinear connections on the 3:1 interaction between NNM6 and NNM17

NC2-Z (\(+,-\)) | NC3-Z (\(+,-\)) | NC4-Z (\(+,-\)) | NC3-Y (\(+,-\)) | NC4-Y (\(+,-\)) | |
---|---|---|---|---|---|

A | (0.2.2) | (0,0) | (0,0) | (0,0) | (0,0) |

B | (0,2.2) | (0,0) | (0,0) | (0,0) | (0,0) |

C | (0,2.2) | (0,0.07) | (0,0) | (0,0) | (0,0) |

D | (0,2.3) | (0,0.19) | (0,0) | (1.1,1.1) | (1.1,1.1) |

E | (0,0) | (0,0) | (0,2.9) | (3.4,3.6) | (3.4,3.4) |

F | (0,0) | (0,0) | (0,0) | (3,3) | (3,3) |

G | (0,0) | (0,0) | (0,0) | (4,4) | (4,4) |

H | (0,0) | (0,0) | (0,2.7) | (3.5,3.8) | (3.7,3.7) |

I | (0,2.1) | (0,0) | (0,1.0) | (0,0) | (0,0) |

J | (0,2.3) | (0,0) | (0,0) | (0,0) | (0,0) |

## 6 Conclusions

The objective of this paper was to investigate the dynamics of a real-life aerospace structure with a strongly nonlinear component. Due to the presence of multiple nonsmooth nonlinearities, closely spaced modes and relatively high damping, this application example poses several challenges. The advanced simulations carried out using numerical continuation showed that the satellite can exhibit a wide variety of nonlinear phenomena including jumps, rich frequency content, quasiperiodic motion, energy transfers from local to global structural modes, internal resonance branches with nonconventional topology, and mode localization. One specific contribution of this work is that several interactions between modes with noncommensurable linear frequencies, observed experimentally, were reproduced with great fidelity using numerical experiments. Overall, a very good qualitative agreement with the results of [15] was obtained. This demonstrates that there now exist in the technical literature effective and rigorous numerical and experimental methods for the analysis of complex, nonlinear industrial structures.

Finally, it is worth noting that the NNMs of the conservative system proved useful to interpret the modal interactions of the real structure. Future investigations should study the influence of damping on the results using, for instance, the concept of NNMs defined as two-dimensional invariant manifolds. Different tools for their computation were recently developed in [35, 36].

## Footnotes

- 1.
This is relevant, because a periodic solution of period \(T\) is periodic with period \(3T\).

## Notes

### Acknowledgments

This paper was prepared in the framework of the European Space Agency (ESA) Technology Research Programme study “Advancement of Mechanical Verification Methods for Nonlinear Spacecraft Structures (NOLISS)” (ESA contract No. 21359/08/NL/SFe). The authors also thank LMS-SAMTECH for providing access to the SAMCEF finite element software. The authors L. Renson and J. P. Noël are Research Fellows (FRIA fellowship) of the *Fonds de la Recherche Scientifique—FNRS*, which is gratefully acknowledged.

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