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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.Comparison of ANM and Predictor-Corrector Method to Continue Solutions of Harmonic Balance Equations
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.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.Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes
http://hdl.handle.net/10985/24164
Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes
DEBEURRE, Marielle; GROLET, Aurélien; THOMAS, Olivier
In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.
Thu, 15 Jun 2023 00:00:00 GMThttp://hdl.handle.net/10985/241642023-06-15T00:00:00ZDEBEURRE, MarielleGROLET, AurélienTHOMAS, OlivierIn this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically-oriented cantilever beam is investigated. The extreme nonlinear vibrations are modeled using a finite element discretization of the geometrically exact beam model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions. The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross-sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration on the natural (linear) modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.Dynamic stability of centrifugal pendulum vibration absorbers allowing a rotational mobility
http://hdl.handle.net/10985/22538
Dynamic stability of centrifugal pendulum vibration absorbers allowing a rotational mobility
MAHE, V.; RENAULT, Alexandre; GROLET, Aurélien; THOMAS, Olivier; MAHE, Hervé
Centrifugal pendulum vibration absorbers (CPVA) are used in the automobile industry to reduce the vibrations of the transmission system. These passive devices are made of several masses oscillating along a given trajectory relative to the rotor. In this paper, the dynamic stability of a new class of CPVA is investigated. The particularity of this new class is that masses now admit a
significant rotation motion relative to the rotor, in addition to the traditional translation motion. The efficiency of such devices is optimal for a perfect synchronous motion of the oscillating masses. However, masses unison can be broken for the benefit of energy localisation on a given absorber, leading to a loss of mitigation performances. To assess the stability of such devices,
a dynamical model based on an analytic perturbation method is established. The aim of this model is to predict analytically localisation and jumps of the response. The validity of the model is confirmed through a comparison with both a numerical resolution of the system’s dynamics and an experimental study.
Sat, 01 Jan 2022 00:00:00 GMThttp://hdl.handle.net/10985/225382022-01-01T00:00:00ZMAHE, V.RENAULT, AlexandreGROLET, AurélienTHOMAS, OlivierMAHE, HervéCentrifugal pendulum vibration absorbers (CPVA) are used in the automobile industry to reduce the vibrations of the transmission system. These passive devices are made of several masses oscillating along a given trajectory relative to the rotor. In this paper, the dynamic stability of a new class of CPVA is investigated. The particularity of this new class is that masses now admit a
significant rotation motion relative to the rotor, in addition to the traditional translation motion. The efficiency of such devices is optimal for a perfect synchronous motion of the oscillating masses. However, masses unison can be broken for the benefit of energy localisation on a given absorber, leading to a loss of mitigation performances. To assess the stability of such devices,
a dynamical model based on an analytic perturbation method is established. The aim of this model is to predict analytically localisation and jumps of the response. The validity of the model is confirmed through a comparison with both a numerical resolution of the system’s dynamics and an experimental study.Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes
http://hdl.handle.net/10985/24781
Extreme nonlinear dynamics of cantilever beams: effect of gravity and slenderness on the nonlinear modes
DEBEURRE, Marielle; GROLET, Aurélien; THOMAS, Olivier
In this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically oriented cantilever beam is investigated. The extreme nonlinear vibrations aremodeled using a finite element discretization of the geometrically exact beam
model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions.The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity
parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration
on the natural (linear)modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.
Thu, 01 Jun 2023 00:00:00 GMThttp://hdl.handle.net/10985/247812023-06-01T00:00:00ZDEBEURRE, MarielleGROLET, AurélienTHOMAS, OlivierIn this paper, the effect of gravity on the nonlinear extreme amplitude vibrations of a slender, vertically oriented cantilever beam is investigated. The extreme nonlinear vibrations aremodeled using a finite element discretization of the geometrically exact beam
model solved in the frequency domain through a combination of harmonic balance and a continuation method for periodic solutions.The geometrically exact model is ideal for dynamic simulations at extreme amplitudes as there is no limitation on the rotation of the cross sections due to the terms governing the rotation being kept exact. It is shown that the very large amplitude vibrations of dimensionless beam structures depend principally on two parameters, a geometrical parameter and a gravity
parameter. By varying these two parameters, the effect of gravity in either a standing or hanging configuration
on the natural (linear)modes as well as on the nonlinear modes in extreme amplitude vibration is studied. It is shown that gravity, in the case of a standing cantilever, is responsible for a linear softening behavior and a nonlinear hardening behavior, particularly pronounced on the first bending mode. These behaviors are reversed for a hanging cantilever.On the dynamic stability and efficiency of centrifugal pendulum vibration absorbers with rotating pendulums
http://hdl.handle.net/10985/22536
On the dynamic stability and efficiency of centrifugal pendulum vibration absorbers with rotating pendulums
MAHÉ, V.; RENAULT, Alexandre; GROLET, Aurélien; MAHÉ, Hervé; THOMAS, Olivier
The automotive industry uses centrifugal pendulum vibration absorbers (CPVAs) to reduce vibrations of the transmission system. These passive devices are made of several masses oscillating along a given path relative to a rotor. This work addresses a recent design of CPVA, in which the pendulums are allowed to rotate relatively to the rotor. The dynamic stability of this CPVA and the shifting of its operating point are investigated in this paper. These two aspects, crucial for an optimal vibration reduction, are assessed using an analytic dynamical model based on a perturbation method. The results obtained allow to propose new design guidelines. The validity of the model is confirmed through a comparison with a numerical resolution of the
system’s dynamics.
Sat, 01 Oct 2022 00:00:00 GMThttp://hdl.handle.net/10985/225362022-10-01T00:00:00ZMAHÉ, V.RENAULT, AlexandreGROLET, AurélienMAHÉ, HervéTHOMAS, OlivierThe automotive industry uses centrifugal pendulum vibration absorbers (CPVAs) to reduce vibrations of the transmission system. These passive devices are made of several masses oscillating along a given path relative to a rotor. This work addresses a recent design of CPVA, in which the pendulums are allowed to rotate relatively to the rotor. The dynamic stability of this CPVA and the shifting of its operating point are investigated in this paper. These two aspects, crucial for an optimal vibration reduction, are assessed using an analytic dynamical model based on a perturbation method. The results obtained allow to propose new design guidelines. The validity of the model is confirmed through a comparison with a numerical resolution of the
system’s dynamics.A comparison of robustness and performance of linear and nonlinear Lanchester dampers
http://hdl.handle.net/10985/22696
A comparison of robustness and performance of linear and nonlinear Lanchester dampers
VAKILINEJAD, Mohammad; GROLET, Aurélien; THOMAS, Olivier
In this paper, we study and compare performance and robustness of linear and nonlinear Lanchester dampers. The linear Lanchester damper consists of a small mass attached to a primary system through a linear dashpot, whereas the nonlinear Lanchester damper is linked to the primary mass through dry friction forces. In each case, we propose a semi-analytical
method for computing the frequency response, for different values of the design parameters, in order to evaluate the performance and robustness of the two kinds of damper. Overall, it is shown that linear Lanchester dampers perform better than nonlinear damper both in terms of attenuation and robustness. Moreover, the nonlinear frequency response curves, that include the
intrinsic non-smooth nature of the friction force, may serve as reference curve for further numerical studies.
Sat, 01 Feb 2020 00:00:00 GMThttp://hdl.handle.net/10985/226962020-02-01T00:00:00ZVAKILINEJAD, MohammadGROLET, AurélienTHOMAS, OlivierIn this paper, we study and compare performance and robustness of linear and nonlinear Lanchester dampers. The linear Lanchester damper consists of a small mass attached to a primary system through a linear dashpot, whereas the nonlinear Lanchester damper is linked to the primary mass through dry friction forces. In each case, we propose a semi-analytical
method for computing the frequency response, for different values of the design parameters, in order to evaluate the performance and robustness of the two kinds of damper. Overall, it is shown that linear Lanchester dampers perform better than nonlinear damper both in terms of attenuation and robustness. Moreover, the nonlinear frequency response curves, that include the
intrinsic non-smooth nature of the friction force, may serve as reference curve for further numerical studies.Subharmonic centrifugal pendulum vibration absorbers allowing a rotational mobility
http://hdl.handle.net/10985/22535
Subharmonic centrifugal pendulum vibration absorbers allowing a rotational mobility
MAHE, V.; RENAULT, Alexandre; GROLET, Aurélien; MAHE, Hervé; THOMAS, Olivier
Rotating machines are often subjected to fluctuating torques, leading to vibrations of the rotor and finally to premature fatigue and noise pollution. This work addresses a new design of centrifugal pendulum vibration absorbers (CPVAs), used to reduce the vibrations in an automotive transmission line. These passive devices, composed of several masses oscillating along a trajectory relative to the rotor, are here tuned at a subharmonic of the targeted harmonic torque frequency. Thanks to the inherent non-linearities, a CPVA with two masses oscillating in phase opposition is able to efficiently counteract the input torque, with particular features such as saturation phenomena. This work particularly extends previous works to a new class of CPVA, whose peculiarity is that masses admit a significant rotation motion relative to the rotor, thus adding the benefit of their rotatory inertia. Results on the system’s subharmonic response and its stability are obtained thanks to an analytical perturbation method, and design
guidelines are proposed. The validity of those results is also confirmed through comparisons with numerical solutions and the performance of this subharmonic system is compared to that of a classical CPVA tuned at the torque frequency.
Thu, 01 Sep 2022 00:00:00 GMThttp://hdl.handle.net/10985/225352022-09-01T00:00:00ZMAHE, V.RENAULT, AlexandreGROLET, AurélienMAHE, HervéTHOMAS, OlivierRotating machines are often subjected to fluctuating torques, leading to vibrations of the rotor and finally to premature fatigue and noise pollution. This work addresses a new design of centrifugal pendulum vibration absorbers (CPVAs), used to reduce the vibrations in an automotive transmission line. These passive devices, composed of several masses oscillating along a trajectory relative to the rotor, are here tuned at a subharmonic of the targeted harmonic torque frequency. Thanks to the inherent non-linearities, a CPVA with two masses oscillating in phase opposition is able to efficiently counteract the input torque, with particular features such as saturation phenomena. This work particularly extends previous works to a new class of CPVA, whose peculiarity is that masses admit a significant rotation motion relative to the rotor, thus adding the benefit of their rotatory inertia. Results on the system’s subharmonic response and its stability are obtained thanks to an analytical perturbation method, and design
guidelines are proposed. The validity of those results is also confirmed through comparisons with numerical solutions and the performance of this subharmonic system is compared to that of a classical CPVA tuned at the torque frequency.