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http://hdl.handle.net/10985/16777
Comparison of ANM and Predictor-Corrector Method to Continue Solutions of Harmonic Balance Equations
WOIWODE, Lukas; BALAJI, Nidish Narayanaa; KAPPAUF, Jonas; TUBITA, Fabia; GUILLOT, Louis; VERGEZ, Christophe; COCHELIN, Bruno; GROLET, Aurélien; KRACK, Malte
In this work we apply and compare two numerical path continuation algorithms for solving algebraic equations arising when applying the Harmonic Balance Method to compute periodic regimes of nonlinear dynamical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach. This algorithm can be applied directly also to non-analytic nonlinearities. The second algorithm relies on a high-order Taylor series expansion of the solution path (the so-called Asymptotic Numerical Method) and can be formulated entirely in the frequency domain. The series expansion can be viewed as a high-order predictor equipped with inherent error estimation capabilities, which permits to avoid correction steps. The second algorithm is limited to analytic nonlinearities, and typically additional variables need to be introduced to cast the equation system into a form that permits the efficient computation of the required high-order derivatives. We apply the algorithms to selected vibration problems involving mechanical systems with polynomial stiffness, dry friction and unilateral contact nonlinearities. We assess the influence of the algorithmic parameters of both methods to draw a picture of their differences and similarities. We analyze the computational performance in detail, to identify bottlenecks of the two methods.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/167772019-01-01T00:00:00ZWOIWODE, LukasBALAJI, Nidish NarayanaaKAPPAUF, JonasTUBITA, FabiaGUILLOT, LouisVERGEZ, ChristopheCOCHELIN, BrunoGROLET, AurélienKRACK, MalteIn this work we apply and compare two numerical path continuation algorithms for solving algebraic equations arising when applying the Harmonic Balance Method to compute periodic regimes of nonlinear dynamical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach. This algorithm can be applied directly also to non-analytic nonlinearities. The second algorithm relies on a high-order Taylor series expansion of the solution path (the so-called Asymptotic Numerical Method) and can be formulated entirely in the frequency domain. The series expansion can be viewed as a high-order predictor equipped with inherent error estimation capabilities, which permits to avoid correction steps. The second algorithm is limited to analytic nonlinearities, and typically additional variables need to be introduced to cast the equation system into a form that permits the efficient computation of the required high-order derivatives. We apply the algorithms to selected vibration problems involving mechanical systems with polynomial stiffness, dry friction and unilateral contact nonlinearities. We assess the influence of the algorithmic parameters of both methods to draw a picture of their differences and similarities. We analyze the computational performance in detail, to identify bottlenecks of the two methods.On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models
http://hdl.handle.net/10985/16775
On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models
GIVOIS, Arthur; GROLET, Aurélien; THOMAS, Olivier; DEÜ, Jean-François
This paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/167752019-01-01T00:00:00ZGIVOIS, ArthurGROLET, AurélienTHOMAS, OlivierDEÜ, Jean-FrançoisThis paper presents a general methodology to compute nonlinear frequency responses of flat structures subjected to large amplitude transverse vibrations, within a finite element context. A reduced-order model (ROM)is obtained by an expansion onto the eigenmode basis of the associated linearized problem, including transverse and in-plane modes. The coefficients of the nonlinear terms of the ROM are computed thanks to a non-intrusive method, using any existing nonlinear finite element code. The direct comparison to analytical models of beams and plates proves that a lot of coefficients can be neglected and that the in-plane motion can be condensed to the transverse motion, thus giving generic rules to simplify theROM. Then, a continuation technique, based on an asymptotic numerical method and the harmonic balance method, is used to compute the frequency response in free (nonlinear mode computation) or harmonically forced vibrations. The whole procedure is tested on a straight beam, a clamped circular plate and a free perforated plate for which some nonlinear modes are computed, including internal resonances. The convergence with harmonic numbers and oscillators is investigated. It shows that keeping a few of them is sufficient in a range of displacements corresponding to the order of the structure’s thickness, with a complexity of the simulated nonlinear phenomena that increase very fast with the number of harmonics and oscillators.Dissipative solitons in forced cyclic and symmetric structures
http://hdl.handle.net/10985/16778
Dissipative solitons in forced cyclic and symmetric structures
HOFFMANN, N.; FONTANELA, Francesco; GROLET, Aurélien; SALLES, Loïc; CHABCHOUB, Amin; CHAMPNEYS, Alan; PATSIAS, Sophoclis; HOFFMANN, Norbert
The emergence of localised vibrations in cyclic and symmetric rotating struc-tures, such as bladed disks of aircraft engines, has challenged engineers in thepast few decades. In the linear regime, localised states may arise due to alack of symmetry, as for example induced by inhomogeneities. However, whenstructures deviate from the linear behaviour, e.g. due to material nonlinearities,geometric nonlinearities like large deformations, or other nonlinear elements likejoints or friction interfaces, localised states may arise even in perfectly symmet-ric structures. In this paper, a system consisting of coupled Duffing oscillatorswith linear viscous damping is subjected to external travelling wave forcing.The system may be considered a minimal model for bladed disks in turboma-chinery operating in the nonlinear regime, where such excitation may arise dueto imbalance or aerodynamic excitation. We demonstrate that near the reso-nance, in this non-conservative regime, localised vibration states bifurcate fromthe travelling waves. Complex bifurcation diagrams result, comprising stableand unstable dissipative solitons. The localised solutions can also be continuednumerically to a conservative limit, where solitons bifurcate from the backbonecurves of the travelling waves at finite amplitudes.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/167782019-01-01T00:00:00ZHOFFMANN, N.FONTANELA, FrancescoGROLET, AurélienSALLES, LoïcCHABCHOUB, AminCHAMPNEYS, AlanPATSIAS, SophoclisHOFFMANN, NorbertThe emergence of localised vibrations in cyclic and symmetric rotating struc-tures, such as bladed disks of aircraft engines, has challenged engineers in thepast few decades. In the linear regime, localised states may arise due to alack of symmetry, as for example induced by inhomogeneities. However, whenstructures deviate from the linear behaviour, e.g. due to material nonlinearities,geometric nonlinearities like large deformations, or other nonlinear elements likejoints or friction interfaces, localised states may arise even in perfectly symmet-ric structures. In this paper, a system consisting of coupled Duffing oscillatorswith linear viscous damping is subjected to external travelling wave forcing.The system may be considered a minimal model for bladed disks in turboma-chinery operating in the nonlinear regime, where such excitation may arise dueto imbalance or aerodynamic excitation. We demonstrate that near the reso-nance, in this non-conservative regime, localised vibration states bifurcate fromthe travelling waves. Complex bifurcation diagrams result, comprising stableand unstable dissipative solitons. The localised solutions can also be continuednumerically to a conservative limit, where solitons bifurcate from the backbonecurves of the travelling waves at finite amplitudes.Computation of quasi-periodic localised vibrations in nonlinear cyclic and symmetric structures using harmonic balance methods
http://hdl.handle.net/10985/16780
Computation of quasi-periodic localised vibrations in nonlinear cyclic and symmetric structures using harmonic balance methods
FONTANELA, Filipe; GROLET, Aurélien; SALLES, Loïc; HOFFMANN, Norbert
In this paper we develop a fully numerical approach to compute quasi-periodic vibrations bifurcating from nonlinear periodic states in cyclic and symmetric structures. The focus is on localised oscillations arising from modulationally unstable travelling waves induced by strong external excitations. The computational strategy is based on the periodic and quasi-periodic harmonic balance methods together with an arc-length continuation scheme. Due to the presence of multiple localised states, a new method to switch from periodic to quasi-periodic states is proposed. The algorithm is applied to two different minimal models for bladed disks vibrating in large amplitudes regimes. In the first case, each sector of the bladed disk is modelled by a single degree of freedom, while in the second application a second degree of freedom is included to account for the disk inertia. In both cases the algorithm has identified and tracked multiple quasi-periodic localised states travelling around the structure in the form of dissipative solitons
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/167802019-01-01T00:00:00ZFONTANELA, FilipeGROLET, AurélienSALLES, LoïcHOFFMANN, NorbertIn this paper we develop a fully numerical approach to compute quasi-periodic vibrations bifurcating from nonlinear periodic states in cyclic and symmetric structures. The focus is on localised oscillations arising from modulationally unstable travelling waves induced by strong external excitations. The computational strategy is based on the periodic and quasi-periodic harmonic balance methods together with an arc-length continuation scheme. Due to the presence of multiple localised states, a new method to switch from periodic to quasi-periodic states is proposed. The algorithm is applied to two different minimal models for bladed disks vibrating in large amplitudes regimes. In the first case, each sector of the bladed disk is modelled by a single degree of freedom, while in the second application a second degree of freedom is included to account for the disk inertia. In both cases the algorithm has identified and tracked multiple quasi-periodic localised states travelling around the structure in the form of dissipative solitonsMultistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators
http://hdl.handle.net/10985/16779
Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators
PAPANGELO, Antonio; FONTANELA, Filipe; GROLET, Aurélien; CIAVARELLA, Michele; HOFFMANN, Norbert
Many engineering structures are composed of weakly coupled sectors assembled in a cyclic and ideally symmetric configuration, which can be simplified as forced Duffing oscillators. In this paper, we study the emergence of localized states in the weakly nonlinear regime. We show that multiple spatially localized solutions may exist, and the resulting bifurcation diagram strongly resembles the snaking pattern observed in a variety of fields in physics, such as optics and fluid dynamics. Moreover, in the transition from the linear to the nonlinear behaviour isolated branches of solutions are identified. Localization is caused by the hardening effect introduced by the nonlinear stiffness, and occurs at large excitation levels. Contrary to the case of mistuning, the presented localization mechanism is triggered by the nonlinearities and arises in perfectly homogeneous systems.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10985/167792019-01-01T00:00:00ZPAPANGELO, AntonioFONTANELA, FilipeGROLET, AurélienCIAVARELLA, MicheleHOFFMANN, NorbertMany engineering structures are composed of weakly coupled sectors assembled in a cyclic and ideally symmetric configuration, which can be simplified as forced Duffing oscillators. In this paper, we study the emergence of localized states in the weakly nonlinear regime. We show that multiple spatially localized solutions may exist, and the resulting bifurcation diagram strongly resembles the snaking pattern observed in a variety of fields in physics, such as optics and fluid dynamics. Moreover, in the transition from the linear to the nonlinear behaviour isolated branches of solutions are identified. Localization is caused by the hardening effect introduced by the nonlinear stiffness, and occurs at large excitation levels. Contrary to the case of mistuning, the presented localization mechanism is triggered by the nonlinearities and arises in perfectly homogeneous systems.