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The present study focuses on the model reduction of non-linear systems. The proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement. These eigenmodes, called the proper orthogonal modes, are optimal with respect to energy content and are used to build a low-dimensional model of the non-linear system. For this purpose, the proper orthogonal modes obtained from a chaotic orbit are considered. Indeed, such an orbit is assumed to cover a portion of the phase space of higher dimension, and hence of greater measure. This higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower-dimensional periodic orbit. In an example, it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non-chaotic response. This is applied to a buckled beam with two permanent magnets and the reduced-order model is validated using both qualitative and quantitative comparisons.

In many domains of applied sciences and in structural dynamics particularly, dealing with large-scale dynamical structures is a central issue. In the presence of non-linearities, seeking for the solution by use of mathematical modelling and simulation (e.g., finite element method) may be computationally intensive. Accordingly, due to the complexity of such a numerical approach, it is worth reducing the dimensionality of the system while retaining its intrinsic properties. The general philosophy of model reduction is to find a co-ordinate transformation in order to sort the components in terms of their influence on the system behaviour. Then, the components of the transformed system with relatively small influence may be truncated without substantially degrading the predictive capability of the model. The proper orthogonal decomposition (POD), also known as Karhunen–Loève transform or principal component analysis (PCA), enables such a co-ordinate transformation. It is a statistical pattern analysis technique for finding the dominant structures in an ensemble of spatially distributed data. These structures, called the proper orthogonal modes (POMs), may be exploited as an orthogonal basis for efficient representation of the ensemble. A key advantage of the decomposition is that each POM is associated with a proper orthogonal value (POV) which provides the relative energy captured by the corresponding mode. Thus, it serves as a well-defined measure of a mode influence on the system behaviour. The present study is motivated by the fact that, in the field of non-linear systems, new features must be defined because mode shapes are no longer effective to represent the system dynamics. While some similarities between the POMs and the modes shapes have been noticed [1] and [2], the POMs are much more useful for capturing the dynamics of a non-linear system. In Ref. [3], lower-dimensional models of non-linear vibrating systems are created using the POMs in order to prove their efficiency and their superiority over the mode shapes. In this work, the POMs obtained from a chaotic orbit are considered. Indeed, such an orbit is assumed to cover a portion of the phase space of higher dimension, and hence of greater measure. This higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower-dimensional periodic orbit. Additionally, if the dimension of the data is d, then the number of states needed to describe the data is bounded by 2d+1, and also leads to a higher number of identifiable modes. The correlation between the dimension of the active phase space and the number of significant proper orthogonal modes has been observed (e.g., in [4]). In the example studied here, it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non-chaotic response. This is applied to a buckled beam with two permanent magnets and the reduced-order model is validated using both qualitative and quantitative comparisons.

In this paper, non-linear systems have been analysed through projection of the equations of motion onto the modes obtained from the POD instead of the modes shapes of the linearised system. The POMs of a chaotic orbit have been considered since these modes better capture the system dynamics. Although it is difficult to draw firm conclusions from a single numerical application, appreciable results have been obtained with the reduced-order model built with these POMs. Furthermore, significant improvements have been brought in comparison with a model built with the modes of a non-chaotic orbit.