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http://hdl.handle.net/10985/8952
Nonlinear forced vibrations of thin structures with tuned eigenfrequencies: the cases of 1:2:4 and 1:2:2 internal resonances
MONTEIL, Mélodie; TOUZÉ, Cyril; THOMAS, Olivier; BENACCHIO, Simon
This paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω1, ω2, and ω3 such that ω3 = 2ω2 = 4ω1, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω3 ω2 2ω1. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamics
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/10985/89522014-01-01T00:00:00ZMONTEIL, MélodieTOUZÉ, CyrilTHOMAS, OlivierBENACCHIO, SimonThis paper is devoted to the analysis of nonlinear forced vibrations of two particular three degrees-of-freedom (dofs) systems exhibiting second order internal resonances resulting from a harmonic tuning of their natural frequencies. The first model considers three modes with eigenfrequencies ω1, ω2, and ω3 such that ω3 = 2ω2 = 4ω1, thus displaying a 1:2:4 internal resonance. The second system exhibits a 1:2:2 internal resonance, so that the frequency relationship reads ω3 ω2 2ω1. Multiple scales method is used to solve analytically the forced oscillations for the two models excited on each degree of freedom at primary resonance. A thorough analytical study is proposed, with a particular emphasis on the stability of the solutions. Parametric investigations allow to get a complete picture of the dynamics of the two systems. Results are systematically compared to the classical 1:2 resonance, in order to understand how the presence of a third oscillator modifies the nonlinear dynamicsEffect of dry friction on a parametrically excited nonlinear oscillator
http://hdl.handle.net/10985/22381
Effect of dry friction on a parametrically excited nonlinear oscillator
THOMAS, Olivier; BENACCHIO, Simon; GIRAUD-AUDINE, Christophe
This study proposes to investigate the effects of dry friction on the behaviour of a parametrically excited nonlinear oscillator using a pendulum as example. A harmonic balance method and time integration simulations are used to respectively compute and validate the solutions of the problem and their stability. The effects of dry friction on the behaviour of the system are discussed.
Sun, 17 Jul 2022 00:00:00 GMThttp://hdl.handle.net/10985/223812022-07-17T00:00:00ZTHOMAS, OlivierBENACCHIO, SimonGIRAUD-AUDINE, ChristopheThis study proposes to investigate the effects of dry friction on the behaviour of a parametrically excited nonlinear oscillator using a pendulum as example. A harmonic balance method and time integration simulations are used to respectively compute and validate the solutions of the problem and their stability. The effects of dry friction on the behaviour of the system are discussed.Effect of dry friction on a parametric nonlinear oscillator
http://hdl.handle.net/10985/22379
Effect of dry friction on a parametric nonlinear oscillator
BENACCHIO, Simon; THOMAS, Olivier; GIRAUD-AUDINE, Christophe
Parametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the method of varying amplitudes are used to find the solutions of this equation and their stability. These results are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the basins of attractions of the solutions are also studied using these simulations. It is found that trivial and nontrivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches. Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.
Wed, 02 Feb 2022 00:00:00 GMThttp://hdl.handle.net/10985/223792022-02-02T00:00:00ZBENACCHIO, SimonTHOMAS, OlivierGIRAUD-AUDINE, ChristopheParametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the method of varying amplitudes are used to find the solutions of this equation and their stability. These results are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the basins of attractions of the solutions are also studied using these simulations. It is found that trivial and nontrivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches. Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.Effect of dry friction on a parametric nonlinear oscillator
http://hdl.handle.net/10985/22705
Effect of dry friction on a parametric nonlinear oscillator
THOMAS, Olivier; BENACCHIO, Simon; GIRAUD-AUDINE, Christophe
Parametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according
to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the
method of varying amplitudes are used to find the solutions of this equation and their stability. These results
are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the
basins of attractions of the solutions are also studied using these simulations. It is found that trivial and non-trivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches.Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.
Tue, 01 Feb 2022 00:00:00 GMThttp://hdl.handle.net/10985/227052022-02-01T00:00:00ZTHOMAS, OlivierBENACCHIO, SimonGIRAUD-AUDINE, ChristopheParametrically excited oscillators are used in several domains, in particular to improve the dynamical behaviour of systems like in the case of the parametric amplification or parametric energy harvesting. Although dry friction is often omitted during system modelling due to the complexity of its nonsmooth nature, it is sometimes necessary to account for this kind of damping to adequately represent the system motion. In this paper, it is proposed to investigate the effect of dry friction on the dynamical behaviour of a nonlinear parametric oscillator. Using the pendulum case as example, the problem is formulated according
to a Mathieu-Duffing equation. Semi-analytical developments using the harmonic balance method and the
method of varying amplitudes are used to find the solutions of this equation and their stability. These results
are validated thanks to a comparison with time integration simulations. Effects of initial conditions on the
basins of attractions of the solutions are also studied using these simulations. It is found that trivial and non-trivial solutions of the oscillator including dry friction are not connected, giving birth to isolated periodic solutions branches.Thus, both initial displacement and phase between the excitation and the oscillator displacement must be carefully chosen to reach periodic solutions. Finally, a method based on the energy principle is used to find the critical forcing amplitude and frequency needed to obtain the birth of nontrivial solutions for the nonlinear parametric oscillator including dry friction.Phase resonance testing of highly flexible structures: Measurement of conservative nonlinear modes and nonlinear damping identification
http://hdl.handle.net/10985/25349
Phase resonance testing of highly flexible structures: Measurement of conservative nonlinear modes and nonlinear damping identification
DEBEURRE, Marielle; BENACCHIO, Simon; GROLET, Aurélien; GRENAT, Clément; GIRAUD-AUDINE, Christophe; THOMAS, Olivier
This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude.
Sat, 01 Jun 2024 00:00:00 GMThttp://hdl.handle.net/10985/253492024-06-01T00:00:00ZDEBEURRE, MarielleBENACCHIO, SimonGROLET, AurélienGRENAT, ClémentGIRAUD-AUDINE, ChristopheTHOMAS, OlivierThis article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude.