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The aim of this contribution is to present numerical comparisons of model-order reduction methods for geometrically nonlinear structures in the general framework of finite element (FE) procedures. Three different methods are compared: the implicit condensation and expansion (ICE), the quadratic manifold computed from modal derivatives (MD), and the direct normal form (DNF) procedure, the latter expressing the reduced dynamics in an invariant-based span of the phase space. The methods are first presented in order to underline their common points and differences, highlighting in particular that ICE and MD use reduction subspaces that are not invariant. A simple analytical example is then used in order to analyze how the different treatments of quadratic nonlinearities by the three methods can affect the predictions. Finally, three beam examples are used to emphasize the ability of the methods to handle curvature (on a curved beam), 1:1 internal resonance (on a clamped-clamped beam with two polarizations), and inertia nonlinearity (on a cantilever beam).

Identification of dissimilatory nitrate reduction to ammonium (DNRA) and denitrification in the dynamic cake layer of a full-scale anoixc dynamic membrane bioreactor (AnDMBR) for treating hotel laundry wastewater was studied. A series of experiments were conducted to understand the contributions of DNRA and canonical denitrification activities in the dynamic cake layer of the AnDMBR. The dynamic cake layer developed included two phases - a steady transmembrane pressure (TMP) increase at 0.24 kPa/day followed by a sharp TMP jump at 1.26 kPa/day four to five days after the AnDMBR start-up. The nitrogen mass balance results showed that canonical denitrification was predominant during the development of the dynamic cake layer. However, DNRA activity and accumulation of bacteria equipped with a complete DNRA pathway showed a positive correlation to the development of the dynamic cake layer. Our metagenomic analysis identified an approximately 18% of the dynamic cake layer bacterial community has a complete DNRA pathway. Pannonibacter (1%), Thauera (0.8%) and Pseudomonas (3%) contained all genes encoding for funcional enzymes of both DNRA (nitrate reductase and DNRA nitrite reductase) and denitrification (nitrate reductase, nitrous oxide reductase and nitric oxide reductase). No other metagenome-assembled genomes (MAGs) possessed a complete cononical denitrification pathway, indicating food-chain-like interactions of denitrifiers in the dynamic cake layer. We found that COD loading rate could be used to control DNRA and canonical denitrification activities during the dynamic cake layer formation.

Non-intrusive methods have been used since two decades to derive reduced-order models for geometrically nonlinear structures, with a particular emphasis on the so-called STiffness Evaluation Procedure (STEP), relying on the static application of prescribed displacements in a finite-element context. We show that a particularly slow convergence of the modal expansion is observed when applying the method with 3D elements, because of nonlinear couplings occurring with very high frequency modes involving 3D thickness deformations. Focusing on the case of flat structures, we first show by computing all the modes of the structure that a converged solution can be exhibited by using either static condensation or normal form theory. We then show that static modal derivatives provide the same solution with fewer calculations. Finally, we propose a modified STEP, where the prescribed displacements are imposed solely on specific degrees of freedom of the structure, and show that this adjustment also provides efficiently a converged solution., Comment: 6 tables, 14 figures, 27 pages

A comparison between two methods to derive reduced-order models (ROM) for geometrically nonlinear structures is proposed. The implicit condensation and expansion (ICE) method relies on a series of applied static loadings. From this set, a stress manifold is constructed for building the ROM. On the other hand, nonlinear normal modes rely on invariant manifold theory in order to keep the key property of invariance for the reduced subspaces. When the model coefficients are fully known, the ICE method reduces to a static condensation. However, in the framework of finite element discretization, getting all these coefficients is generally too computationally expensive. The stress manifold is shown to tend to the invariant manifold only when a slow/fast decomposition between master and slave coordinates can be assumed. Another key problem in using the ICE method is related to the fitting procedure when a large number of modes need to be taken into account. A simplified procedure, relying on normal form theory and identification of only resonant monomial terms in the nonlinear stiffness, is proposed and contrasted with the current method. All the findings are illustrated on beams and plates examples.

The direct computation of the third-order normal form for a geometrically nonlinear structure discretised with the finite element (FE) method, is detailed. The procedure allows to define a nonlinear mapping in order to derive accurate reduced-order models (ROM) relying on invariant manifold theory. The proposed reduction strategy is direct and simulation free, in the sense that it allows to pass from physical coordinates (FE nodes) to normal coordinates, describing the dynamics in an invariant-based span of the phase space. The number of master modes for the ROM is not a priori limited since a complete change of coordinate is proposed. The underlying theory ensures the quality of the predictions thanks to the invariance property of the reduced subspace, together with their curvatures in phase space that accounts for the nonresonant nonlinear couplings. The method is applied to a beam discretised with 3D elements and shows its ability in recovering internal resonance at high energy. Then a fan blade model is investigated and the correct prediction given by the ROMs are assessed and discussed. A method is proposed to approximate an aggregate value for the damping, that takes into account the damping coefficients of all the slave modes, and also using the Rayleigh damping model as input. Frequency-response curves for the beam and the blades are then exhibited, showing the accuracy of the proposed method., 34 pages, 10 figures, 2 tables, submitted to CMAME

In this paper, a simple and efficient method to suppress excess vibration on an arbitrarily supported rectangular damped plate subjected to steady-state harmonic excitations is developed. This vibration suppression is achieved by enforcing points of zero displacement, or otherwise referred to as nodes, at desired locations on the plate using properly tuned damped vibration absorbers. A systematic design procedure that consists of two steps is introduced to enforce the specified nodes. In the first step, sets of feasible attachment locations for the absorbers are identified, and in the second step, the required absorber parameters that are physically realizable are obtained, under the constraints of a tolerable vibration amplitude for the absorber masses. Numerical experiments show that by inducing nodes at the appropriate locations, points of nearly zero vibration amplitudes can be enforced, effectively quenching vibration in that region of the plate.

In this paper, a simple and effective method to enforce fixed nodes, or points of zero displacement and zero slope, on an arbitrarily supported rectangular plate subjected to steady-state harmonic excitations is developed. This is achieved by attaching properly tuned translational and rotational oscillators at specified locations. The governing equations of the combined system are first derived using the assumed-modes method. By enforcing the conditions of zero displacements and zero slopes simultaneously, a set of constraint equations are formulated, from which the oscillator parameters can be determined. When the attachment locations coincide with the desired fixed node locations, it is always possible to select the oscillator parameters such that one or multiple fixed nodes are induced at any locations on the plate for any excitation frequency. When the attachment and the desired node locations are not collocated, it is only possible to induce nodes at certain locations on the plate. When the fixed node locations are judiciously chosen, a selected region of the plate can be made to remain nearly stationary. Thus, the proposed method provides a simple and yet effective means to passively control excessive vibrations.