16 A. J. Elliott Given their similarity, the comparison of these methodologies has been relatively common [9], with the authors of [10] investigating the impact that their differences have on the accuracy of the associated NIROMs. A key finding of this work was that accurately capturing the cross-coupling of the vibration modes was fundamental to achieving an accurate NIROM, an attribute that was not consistently observed for either method. In light of this underperformance, the recent literature has started to explore the use of machine learning, and particularly recurrent neural networks (RNNs), to overcome this barrier [11–14]. Simpson et al. [11] employ several toy models to investigate the use of long short-term memory (LSTM) RNNs [15], utilising an autoencoder that reduces the order of the model. In contrast, Cenedese et al. [12, 13] expand the spectral submanifold approach to nonlinear normal modes by incorporating a data-driven learning of the dynamics on the submanifold. Across these examples, the methodologies show exceptional promise, though their relative complexity restricts their applicability by a non-expert in a real-world setting. This paper explores the initial steps in the development of an analogous strategy that aims to directly address this need. A simple, five-degree-of-freedom (5DOF) mass-spring model, including nonlinear springs, is introduced to explore the fundamental accuracy and practicability of the proposed methodology. The application of physics-informed (PI) RNNs, in which verifiable knowledge of the physics of the system is used to guarantee convergence to a physically interpretable solution [16], is identified as a key strategy in achieving this overall aim. 3.2 Methodology The general equations of motion for a nonlinear mechanical system are given by .M¨x+Cx˙x+Kx+FNL,x(x) =Fx, (3.1 ) where . x denotes the physical coordinates and. M , . C , an d. Kdenote the mass, damping, and stiffness matrices, respectively. .FNL(x) an d. Fx represent the vectors of nonlinear and external forces; note that the subscript x on the forces and damping matrix denotes the fact that this vector is in physical coordinates, rather than modal, as this will simplify the notation below. It is common practice to consider this system in terms of the basis of modal coordinates—denoted by. q—which can be found through the application of the mass-normalised eigenvector matrix, . , with .x = q . Through this transformation, Eq. (3.1) can be rewritten as [3] . ¨q+C˙q+ q+FNL(q) =F, (3.2 ) where . is the diagonal matrix of the squared natural frequencies (i.e. the nth diagonal entry is . ω 2 n ) an d . FNL(q) = TFNL,x( q) . The NIROM methodologies discussed in the Introduction are designed to preserve the structure of Eq. (3.2), but reducing the number of modes retained in the model. Specifically, the NIROM is defined in terms of some subset .{ˆq} ⊂ {q}, with associated reduced eigenvector matrix. ˆ . The number of modes retained in. {ˆq} will be denoted byR . The reduced equations of motion can now be written in the form . ¨ˆq+ ˆC˙ˆq+ ˆ ˆq+ ˆFNL(ˆq) = ˆF. (3.3 ) Both the linear components of Eq. (3.1) and. ˆ can be readily obtained, either analytically or from the finite element software, so the linear elements of Eq. (3.3) can be easily calculated. In contrast, the nonlinear function is typically complicated and is often approximated through a numerical methodology that is not made available to the user. Thus, the main aim of the NIROM generation strategy is to analytically approximate this behaviour in the function. ˆFNL . Since the motivation for developing NIROMs is frequently driven by a need for significant improvements in response time, . ˆFNL is usually written as a cubic polynomial in terms of the elements of . ˆq. The. mth element of this vector is given by . ˆFNL,m = R i=1 R j=1 R k=1 A(m) i,j,k ˆ qi ˆqj ˆqk + R i=1 R j=1 B (m) i,j ˆ qi ˆqj, (3.4 ) where the coefficients, .Ai,j,k and. Bi,j , are the terms estimated by the reduction steps. With the nonlinear terms approximated as in Eq. (3.4), the static cases applied to the system take the form
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