1,720,978 research outputs found
Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: Comparison of POD and asymptotic nonlinear normal modes methods
No full textModal approach for nonlinear vibrations of damped impacted plates: Application to sound synthesis of gongs and cymbals
No full textThis paper presents a modal, time-domain scheme for the nonlinear vibrations of perfect and imperfect plates. The scheme can take into account a large number of degrees-of-freedom and is energy-conserving. The targeted application is the sound synthesis of cymbals and gong-like musical instruments, which are known for displaying a strongly nonlinear vibrating behaviour. This behaviour is typical of a wave turbulence regime, in which the wide-band spectrum of excited modes is observable in the form of an energy cascade. The modal method is selected for its versatility in handling complex damping laws that can be implemented easily by selecting appropriate damping values in each one of the modal equations. In the first part of the paper, the modal method is explained in its generality, and it will be seen that the method is valid for plates with arbitrary geometry and boundary conditions as long as the eigenmodes are known. Secondly, a time-integration, energy-conserving scheme for perfect and imperfect plates is presented, and implementation comments are given in order to treat efficiently the high-dimensionality of the resulting dynamical system. The scheme is run with appropriate parameters in order to produce sound samples. A simple impact law is considered for the excitation, whereas the flexibility of the method is highlighted by showing simulations for free-edge circular plates and simply-supported rectangular plates, together with various damping laws
Nonlinear model order reduction of resonant piezoelectric micro-actuators: An invariant manifold approach
No full textThis paper presents a novel derivation of the direct parametrisation method for invariant manifolds able to build simulation-free reduced-order models for nonlinear piezoelectric structures, with a particular emphasis on applications to Micro Electro Mechanical Systems. The constitutive model adopted accounts for the hysteretic and electrostrictive response of the piezoelectric material by resorting to the Landau-Devonshire theory of ferroelectrics. Results are validated with full-order simulations operated with a harmonic balance finite element method to highlight the reliability of the proposed reduction procedure. Numerical results show a remarkable gain in terms of computing time as a result of the dimensionality reduction process over low dimensional invariant sets. Results are also compared with experimental data to highlight the remarkable benefits of the proposed model order reduction technique
High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point
Full text linkThis paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems in oscillatory form expressed in the physical basis, so that the technique is directly applicable to mechanical problems discretised by the finite element method. Two nonlinear mappings, respectively related to displacement and velocity, are introduced, and the link between the two is made explicit at arbitrary order of expansion, under the assumption that the damping matrix is diagonalised by the conservative linear eigenvectors. The same development is performed on the reduced-order dynamics which is computed at generic order following different styles of parametrisation. More specifically, three different styles are introduced and commented: the graph style, the complex normal form style and the real normal form style. These developments allow making better connections with earlier works using these parametrisation methods. The technique is then applied to three different examples. A clamped-clamped arch with increasing curvature is first used to show an example of a system with a softening behaviour turning to hardening at larger amplitudes, which can be replicated with a single mode reduction. Secondly, the case of a cantilever beam is investigated. It is shown that invariant manifold of the first mode shows a folding point at large amplitudes. This exemplifies the failure of the graph style due to the folding point on a real structure, whereas the normal form style is able to pass over the folding. Finally, a MEMS (Micro Electro Mechanical System) micromirror undergoing large rotations is used to show the importance of using high-order expansions on an industrial example
Model order reduction based on direct normal form: application to large finite element MEMS structures featuring internal resonance
Full text linkDimensionality reduction in mechanical vibratory systems poses challenges for distributed structures including geometric nonlinearities, mainly because of the lack of invariance of the linear subspaces. A reduction method based on direct normal form computation for large finite element (FE) models is here detailed. The main advantage resides in operating directly from the physical space, hence avoiding the computation of the complete eigenfunctions spectrum. Explicit solutions are given, thus enabling a fully non-intrusive version of the reduction method. The reduced dynamics is obtained from the normal form of the geometrically nonlinear mechanical problem, free of non-resonant monomials, and truncated to the selected master coordinates, thus making a direct link with the parametrisation of invariant manifolds. The method is fully expressed with a complex-valued formalism by detailing the homological equations in a systematic manner, and the link with real-valued expressions is established. A special emphasis is put on the treatment of second-order internal resonances and the specific case of a 1:2 resonance is made explicit. Finally, applications to large-scale models of micro-electro-mechanical structures featuring 1:2 and 1:3 resonances are reported, along with considerations on computational efficiency
High-order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to generic forcing terms and parametrically excited systems
No full textThe direct parametrisation method for invariant manifolds is used for model order reduction of forced-damped mechanical structures subjected to geometric nonlinearities. Nonlinear mappings are introduced, allowing one to pass from the degrees of freedom of the finite-element model to the normal coordinates. Arbitrary orders of expansions are considered for the unknown mappings and the reduced dynamics, which are then solved sequentially through the homological equations for both autonomous and time-dependent terms. It is emphasised that the two problems share a similar structure, which can be used for an efficient implementation of the non-autonomous added terms. Special emphasis is also put on the new resonance conditions arising due to the presence of the external forcing frequencies, which allow predicting phenomena such as parametric excitation and isolas formation. The method is then applied to structures of academic and industrial interest. First, the large amplitude vibrations of a forced-damped cantilever beam are studied. This example highlights that high-order non-autonomous terms are compulsory to correctly estimate the maximum vibration amplitude experienced by the structure. The birth of isolated solutions is also illustrated on this example. The cantilever is then used to show how quadratic coupling creates conditions for the excitation of the parametric instability, and that this feature is correctly embedded in the reduction process. A shallow arch excited with multi-modal forcing is then studied to detail different forcing effects. Finally, the approach is validated on a structure of industrial relevance, i.e. a comb-driven micro-electro-mechanical resonator. The accuracy and computational performance reported suggest that the proposed methodology can accurately predict the nonlinear dynamic response of a large class of nonlinear vibratory systems
Backbone curves, Neimark-Sacker boundaries and appearance of quasi-periodicity in nonlinear oscillators: application to 1:2 internal resonance and frequency combs in MEMS
Full text linkQuasi-periodic solutions can arise in assemblies of nonlinear oscillators as a consequence of Neimark-Sacker bifurcations. In this work, the appearance of Neimark-Sacker bifurcations is investigated analytically and numerically in the specific case of a system of two coupled oscillators featuring a 1:2 internal resonance. More specifically, the locus of Neimark-Sacker points is analytically derived and its evolution with respect to the system parameters is highlighted. The backbone curves, solution of the conservative system, are first investigated, showing in particular the existence of two families of periodic orbits, denoted as parabolic modes. The behaviour of these modes, when the detuning between the eigenfrequencies of the system is varied, is underlined. The non-vanishing limit value, at the origin of one solution family, allows explaining the appearance of isolated solutions for the damped-forced system. The results are then applied to a Micro-Electro-Mechanical System-like shallow arch structure, to show how the analytical expression of the Neimark-Sacker boundary curve can be used for rapid prediction of the appearance of quasiperiodic regime, and thus frequency combs, in Micro-Electro-Mechanical System dynamics
Dynamics of the wave turbulence spectrum in vibrating plates: A numerical investigation using a conservative finite difference scheme
No full textThe dynamics of the local kinetic energy spectrum of an elastic plate vibrating in a wave turbulence (WT) regime is investigated with a finite difference, energy-conserving scheme. The numerical method allows the simulation of pointwise forcing together with realistic boundary conditions, a set-up which is close to experimental conditions. In the absence of damping, the framework of non-stationary wave turbulence is used. Numerical simulations show the presence of a front propagating to high frequencies, leaving a steady spectrum in its wake. Self-similar dynamics of the spectra are found with and without periodic external forcing. For the periodic forcing, the mean injected power is found to be constant, and the frequency at the cascade front evolves linearly with time resulting in a increase of the total energy. For the free turbulence, the energy contained in the cascade remains constant while the frequency front increases as t1/3. These self-similar solutions are found to be in accordance with the kinetic equation derived from the von Kármán plate equations. The effect of the pointwise forcing is observable and introduces a steeper slope at low frequencies, as compared to the unforced case. The presence of a realistic geometric imperfection of the plate is found to have no effect on the global properties of the spectra dynamics. The steeper slope brought by the external forcing is shown to be still observable in a more realistic case where damping is added. © 2014 Elsevier B.V. All rights reserved
Nonlinear dynamics of rectangular plates: Investigation of modal interaction in free and forced vibrations
No full textNonlinear vibrations of thin rectangular plates are considered, using the von kármán equations in order to take into account the effect of geometric nonlinearities. Solutions are derived through an expansion over the linear eigenmodes of the system for both the transverse displacement and the Airy stress function, resulting in a series of coupled oscillators with cubic nonlinearities, where the coupling coefficients are functions of the linear eigenmodes. A general strategy for the calculation of these coefficients is outlined, and the particular case of a simply supported plate with movable edges is thoroughly investigated. To this extent, a numerical method based on a new series expansion is derived to compute the Airy stress function modes, for which an analytical solution is not available. It is shown that this strategy allows the calculation of the nonlinear coupling coefficients with arbitrary precision, and several numerical examples are provided. Symmetry properties are derived to speed up the calculation process and to allow a substantial reduction in memory requirements. Finally, analysis by continuation allows an investigation of the nonlinear dynamics of the first two modes both in the free and forced regimes. Modal interactions through internal resonances are highlighted, and their activation in the forced case is discussed, allowing to compare the nonlinear normal modes (NNMs) of the undamped system with the observable periodic orbits of the forced and damped structure. © 2013 Springer-Verlag Wien
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