Strongly Modulated Response Research Articles

Targeted energy transfer in a 2-DOF mechanical system coupled to a non-linear energy sink with varying stiffness

The dynamical behavior of a non-linear mechanical system with two degrees of freedom (DOFs) during free and forced excitations is studied analytically and numerically. The non-linearity of the system is represented intentionally by a smooth non-linear simple function with periodically varying stiffness around a constant value for the sake of practical investigations. Analysis of the system leads to a method that could be used to design the non-linear energy sink (NES) so that the behavior of the system during relaxation and its strongly modulated response (SMR) could be improved versus the constant stiffness configuration.

Targeted Energy Transfer Under Harmonic Forcing With a Vibro-Impact Nonlinear Energy Sink: Analytical and Experimental Developments

Recently, it has been demonstrated that a vibro-impact type nonlinear energy sink (VI-NES) can be used efficiently to mitigate vibration of a linear oscillator (LO) under transient loading. The objective of this paper is to investigate theoretically and experimentally the potential of a VI-NES to mitigate vibrations of an LO subjected to a harmonic excitation (nevertheless, the presentation of an optimal VI-NES is beyond the scope of this paper). Due to the small mass ratio between the LO and the flying mass of the NES, the obtained equations of motion are analyzed using the method of multiple scales in the case of 1:1 resonance. It is shown that in addition to periodic response, system with VI-NES can exhibit strongly modulated response (SMR). Experimentally, the whole system is embedded on an electrodynamic shaker. The VI-NES is realized with a ball which is free to move in a cavity with a predesigned gap. The mass of the ball is less than 1% of the mass of the LO. The experiment confirms the existence of periodic and SMR regimes. A good agreement between theoretical and experimental results is observed.

Experimental Investigation and Design Optimization of Targeted Energy Transfer Under Periodic Forcing

In this paper, the dynamic response of a harmonically forced linear oscillator (LO) strongly coupled to a nonlinear energy sink (NES) is investigated both theoretically and experimentally. The system studied comprises an LO with an embedded, purely cubic NES. The behavior of the system is analyzed in the vicinity of 1:1 resonance. The complexification-averaging technique is used to obtain modulation equations and the associated fixed points. These modulation equations are analyzed using asymptotic expansion to study the regimes related to relaxation oscillation of the slow flow, called strongly modulated response (SMR). The zones where SMR occurs are computed using a mapping procedure. The slow invariant manifolds (SIM) are used to derive a proper optimization procedure. It is shown that there is an optimal zone in the forcing amplitude-nonlinear stiffness parameter plane, where SMR occurs without having a high amplitude detached resonance tongue. Two experimental setups are presented. One is not optimized and has a relatively high mass ratio (≈13%) and the other one is optimized and exhibits strong mass asymmetry (mass ratio ≈1%). Different frequency response curves and associated zones of SMR are obtained for various forcing amplitudes. The reported experimental results confirm the design procedure and the possible application of NES for vibration mitigation under periodic forcing.

Chatter Control in Turning Process with a Nonlinear Energy Sink

This paper presents the interest of an original absorber of vibration in order to reduce chatter vibration in turning process. The device is composed of a linear oscillator corresponding to a flexible cutting tool subject to chatter strongly coupled to a Nonlinear Energy Sink (NES), with purely cubic stiffness. The novelty of this work is the use of a nonlinear cutting law, more accurate for modeling the cutting process. The delayed equations of motion are analyzed using a combination of the method of multiple scales and harmonic balance. Different types of responses regimes are revealed such as periodic response and also Strongly Modulated Response (SMR). Analytic results are then compared with numerical simulations. Finally, the potential of the NES is demonstrated to control chatter in turning process.

Vibratory energy exchange between a linear and a nonsmooth system in the presence of the gravity

The concept of the vibratory energy transfer between a linear master DOF and a nonsmooth nonlinear energy sink (NES) in the presence of gravity forces is studied. Different invariant manifolds of the system at different time scales are revealed and necessary conditions for leading the behavior of the system to strongly modulated response (SMR) are enlightened.

Delayed dynamical sytem strongly coupled to a nonlinear energy sink : application to machining chatter

This paper investigate the di erent response regimes of a cutting tool on a lathe strongly coupled to a nonlinear energy sink. The equations of motion are analysed via the method of multiple scales. Condition of elimination of secular terms permit to derive equation of the slow invariant manifold (SIM) and the behavior of the system has been explained by studying the location of the fixed points of the slow flow on the SIM. Di erent types of responses are revealed such as periodic response and also strongly modulated response (SMR) wich are not related to the fixed points of the slow flow. Analytic results are then compared to numerical simulations. also benefit for passive control. In this paper, the possi- bility of controlling regenerative chatter using a Nonlinear Energy Sink (NES) is analyzed for turning process. The- oretical predictions are confirmed with numerical integra- tion. The paper is organized as follow. In the next section, the model considered in this study is described. In the third section, the underlying linear sub-system is analyzed. In the fourth section the asymptotic analysis of the equation of motion is performed. Then, di erent response regimes accompanied with numerical simulation are presented.

Response regimes in forced system with non-linear energy sink: quasi-periodic and random forcing

The system under investigation comprises a linear oscillator coupled to a non-linear energy sink (NES) under quasi-periodic forcing in the regime of 1:1:1 resonance. Interaction of the quasi-periodic excitation with the strongly modulated response (SMR) regime is studied in detail both analytically and numerically. Theoretical study developed in the paper allows establishing the threshold value for the amplitude of modulation beyond which SMR regime is excited. This phenomenon is of great practical use since applying the quasi-periodic excitation beyond the threshold results in elimination of possible undesired regimes causing high-amplitude oscillations of the main structure. Bifurcations of the SMR caused by quasi-periodic excitation were analyzed with the help of semi-analytical procedure based on two-dimensional maps. Numerical evidences for exciting the strongly modulated bursts in the response by a random, quasi-periodic narrow-band excitation are also provided. Fairly good correspondence was observed between analytical model and numerical simulations.

Bifurcations of self-excitation regimes in a Van der Pol oscillator with a nonlinear energy sink

The paper investigates regimes of self-excitation in a Van der Pol oscillator with an attached nonlinear energy sink (NES). Initial equations are reduced by averaging to a 3D system. The small relative mass of the NES justifies analysis of this averaged system as singularly perturbed with two “slow” and one “super–slow” variable. Such an approach, in turn, provides a complete analytic description of possible response regimes. In addition to almost unperturbed limit cycle oscillations (LCOs), the system can exhibit complete elimination of self-excitation, small-amplitude LCOs as well as excitation of a quasiperiodic strongly modulated response (SMR). In the space of parameters, the latter can be approached by three distinct bifurcation mechanisms: canard explosion, Shil’nikov bifurcation and heteroclinic bifurcation. Some of the above oscillatory regimes can co-exist for the same values of the system parameters. In this case, it is possible to establish the basins of attraction for the co-existing regimes. Direct numeric simulations demonstrate good coincidence with the analytic predictions.

Response regimes of linear oscillator coupled to nonlinear energy sink with harmonic forcing and frequency detuning

Dynamical system under investigation in the current work is comprised of harmonically forced linear oscillator with attached nonlinear energy sink. External forcing frequency detuning near the main resonance (1:1) is included in the system investigation. The detailed study of the periodic and quasiperiodic regimes is done in the work via (adaptive) averaging method. Local bifurcations of the periodic regimes are revealed and fully described in the space of system parameters (amplitude of excitation, damping, and frequency detuning). Novel analytical approach for predictions of strongly modulated response (SMR) is presented. This approach provides a sufficient condition for the SMR existence contrary to the previous studies. Various possibilities of coexistence of the response regimes are predicted analytically and demonstrated numerically. Among those is a coexistence of two distinct periodic regimes together with the SMR. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.

(P8, R2) Approximate constraints of motion for classically chaotic vibrational dynamics: Vague Tori, semiclassical quantization, and classical intramolecular energy flow

Dynamical system under investigation in the current work is comprised of harmonically forced linear oscillator with attached nonlinear energy sink. External forcing frequency detuning near the main resonance (1:1) is included in the system investigation. The detailed study of the periodic and quasiperiodic regimes is done in the work via (adaptive) averaging method. Local bifurcations of the periodic regimes are revealed and fully described in the space of system parameters (amplitude of excitation, damping, and frequency detuning). Novel analytical approach for predictions of strongly modulated response (SMR) is presented. This approach provides a sufficient condition for the SMR existence contrary to the previous studies. Various possibilities of coexistence of the response regimes are predicted analytically and demonstrated numerically. Among those is a coexistence of two distinct periodic regimes together with the SMR. All findings of the simplified analytic model are verified numerically and considerable agreement is observed.