M. J. Béthenod, Sur l'entretien du mouvement d'un pendule au moyen d'un courant alternatif de fréquenceélevée par rapportà sa fréquence propre. Comptes rendus hebdomadaires de l'Académie des sciences, vol.207, p.847, 1938.

D. Cintra and P. , Argumentary oscillation phenomenon, online publication, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00876552

D. Cintra and P. , Argumentary Duffing oscillators -Stable-regime probability, online publication, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01123884

D. Cintra and P. , Six models of argumental oscillators -Experimental results, 2014.

A. Cornu, Sur la synchronisation des horloges de précision et la distribution de l'heure, Journal de Physique Théorique et Appliquée, vol.7, issue.1, pp.231-239, 1888.

B. Cretin and D. Vernier, Quantized amplitudes in a nonlinear resonant electrical circuit, Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium, vols 1 and 2, vol.1, pp.797-800, 2009.

G. Darrieus, Joseph Béthenod -Sa vie, son oeuvre -Présentationà la Société francaise des Electriciens le 2 décembre 1944, vol.67, 1945.

D. Doubochinski and J. Tennenbaum, On the general nature of physical objects and their interactions, as suggested by the properties of argumentally-coupled oscillating systems, 2008.

D. B. Doubochinski and J. B. Doubochinski, Amorçage argumentaire d'oscillations entretenues avec une série discrète d'amplitudes stables. E.D.F. Bulletin de la direction desétudes et recherches, série C mathématiques, informatique, vol.3, pp.11-20, 1991.

H. , O. Ekici, and H. Boyaci, Effects of non-ideal boundary conditions on vibrations of microbeams, Journal of Vibration and Control, vol.13, issue.9, pp.1369-1378, 2007.

. Ch and . Féry, Sur quelques modesélectriques d'entretien du pendule. Pendule sans lien matériel, Journal de Physique Théorique et Appliquée, vol.7, issue.1, pp.520-530, 1908.

D. I. Penner, D. B. Doubochinski, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Soviet Physics Uspekhi, vol.16, issue.1, 1973.

D. I. Penner, Y. B. Doubochinski, D. B. Doubochinski, and M. I. Kozakov, Oscillations with self-regulating interaction time, Soviet Physics Doklady, vol.17, p.541, 1972.

P. Popovic, A. H. Nayfeh, K. Oh, and S. A. Nayfeh, An experimental investigation of energy transfer from a high-frequency mode to a low-frequency mode in a flexible structure, Journal of Vibration and Control, vol.1, issue.1, pp.115-128, 1995.

J. P. Treilhou, J. Coutelier, J. J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers, Advances in Space Research, vol.26, issue.9, pp.1423-1426, 2000.

N. Bogolioubov and I. Mitropolski, Les méthodes asymptotiques en théorie des oscillations non linéaires. Gauthiers-Villars, 1962.

D. Cintra and P. , Nonlinear argumental oscillators: A few examples of modulation via spatial position, Journal of Vibration and Control, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01252588

D. Cintra and P. Argoul, Attractors capture probability in nonlinear argumental oscillators, Communications in Nonlinear Science and Numerical Simulation, vol.48, pp.150-169, 2017.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition. Alan Jeffrey and Daniel Zwillinger, 2007.

W. Lacarbonara, D. Bernardini, and F. Vestroni, Nonlinear thermomechanical oscillations of shape-memory devices, International Journal of Solids and Structures, vol.41, issue.56, pp.1209-1234, 2004.

D. I. Penner, D. B. Doubochinski, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Soviet Physics Uspekhi, vol.16, issue.1, pp.158-160, 1973.

G. Rega and F. Benedettini, Planar non-linear oscillations of elastic cables under subharmonic resonance conditions, Journal of Sound and Vibration, vol.132, issue.3, pp.367-381, 1989.

J. P. Treilhou, J. Coutelier, J. J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers, Advances in Space Research, vol.26, issue.9, pp.1423-1426, 2000.

M. Béthenod, Sur l'entretien du mouvement d'un pendule au moyen d'un courant alternatif de fréquenceélevée par rapportà sa fréquence propre. Comptes rendus hebdomadaires de l'Académie des sciences, vol.207, pp.847-856, 1938.

D. I. Penner, D. B. Doubochinski, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Sov. Phys.-Usp, vol.16, issue.1, pp.158-60, 1973.

N. Bogolioubov and I. Mitropolski, Les méthodes asymptotiques en théorie des oscillations non linéaires. Gauthiers-Villars, 1962.

D. Doubochinski and J. Doubochinski, Amorçage argumentaire d'oscillations entretenues avec une série discrète d'amplitudes stables. E.D.F. Bulletin de la direction desétudes et recherches, série C mathématiques, informatique, vol.3, pp.11-20, 1991.

D. Cintra and P. Argoul, Nonlinear argumental oscillators: A few examples of modulation via spatial position, J Vib Control, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01252588

D. Cintra and P. Argoul, Nonlinear argumental oscillators: Stability criterion and approximate implicit analytic solution
URL : https://hal.archives-ouvertes.fr/hal-01701277

J. Treilhou, J. Coutelier, J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers, Adv Space Res, vol.26, issue.9, pp.1423-1429, 2000.

B. Cretin and D. Vernier, Quantized amplitudes in a nonlinear resonant electrical circuit, Joint Meeting of the 23rd European Frequency and Time Forum/IEEE International Frequency Control Symposium. Available from, vol.1, pp.797-800, 2009.

L. Abobda and P. Woafo, Subharmonic and bursting oscillations of a ferromagnetic mass fixed on a spring and subjected to an AC electromagnet, Commun Nonlinear SCI, vol.17, issue.7, pp.3082-91, 2012.

J. A. Wright, J. H. Deane, M. Bartuccelli, and G. Gentile, Basins of attraction in forced systems with time-varying dissipation, Commun Nonlinear SCI, vol.29, issue.13, pp.72-87, 2015.

, Equation of motion with the natural model

, Equation of motion with the smooth model

. .. , 11 Decomposing function H(a sin(? )) into a Fourier series of variable ?

. .. Stationary-condition, 12 ? -curve, G-curve and stationary-solutions curve, p.12

. .. Excitation-threshold, 12 Graphic representation of the stationary solutions in the (a S , a A )-plane, p.12

.. .. Stability,

. .. Case-a-a-<-a-acrit, , p.13

. .. Case-a-a->-a-acrit, , p.14

. .. Frequency-response, 1 Smooth model of the external force and averaging method. 16 9.2 Smooth model of the external force and original secondorder equation, vol.16

. .. Conclusion, 17 10 Natural model of the external force, 17 Construction of the stable and unstable stationaryregime representative points, p.17

, Other cases with the smooth model

. .. Model-comparison, , p.19

.. .. Conclusion,

A. Appendix, Approximation to a truncated parabola

. .. Composite-parabola-case, , p.23

, Appendix B: a stability criterion

N. Bogolioubov and I. Mitropolski, Les méthodes asymptotiques en théorie des oscillations non linaires. Gauthiers-Villars, 1962.

D. Cintra and P. Argoul, Attractors capture probability in nonlinear argumental oscillators, Communications in Nonlinear Science and Numerical Simulation, vol.48, pp.150-169, 2017.

D. Cintra and P. Argoul, Non-linear argumental oscillators: Stability criterion and approximate implicit analytic solution, International Journal of Non-Linear Mechanics, vol.94, pp.109-124, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01701277

D. Cintra and P. Argoul, Nonlinear argumental oscillators: A few examples of modulation via spatial position, Journal of Vibration and Control, vol.23, issue.18, pp.2888-2911, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01252588

D. Cintra, G. Cumunel, and P. Argoul, Experimental study of the argumental transverse vibration of a beam excited through permanent or intermittent elastic contact by a harmonic axial motion, Journal to be, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01636638

D. Cintra, G. Cumunel, and P. Argoul, A few properties in symbolic form for the argumental transverse vibration of a beam excited through permanent or intermittent elastic contact by a harmonic axial motion, Journal to be, 2017.

B. Cretin and D. Vernier, Quantized amplitudes in a nonlinear resonant electrical circuit, 2009 Joint Meeting of the European Frequency and Time Forum and the IEEE International Frequency Control Symposium, vols 1 and 2, vol.1, pp.797-800, 2009.

D. Doubochinski, Argumental oscillations. Macroscopic quantum effects, SciTech Library, 2015.

J. Humar, Dynamics of Structures. A.A. Balkema, 2001.

D. I. Penner, D. B. Duboshinskii, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Phys. Usp, vol.16, issue.1, pp.158-160, 1973.

B. Pratiher and S. K. Dwivedy, Nonlinear response of a flexible cartesian manipulator withpayload and pulsating axial force, Nonlinear Dynamics, vol.57, issue.1, pp.177-195, 2009.

J. Treilhou, J. Coutelier, J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers. Advances in, Space Research, vol.26, issue.9, pp.1423-1426, 2000.

D. Cintra, G. Cumunel, and P. Argoul, Modeling and numerical results for the argumental transverse vibration of a beam excited through permanent or intermittent elastic contact by a harmonic axial motion, Journal to be, 2017.

D. Cintra and P. Argoul, Attractors capture probability in nonlinear argumental oscillators, Communications in Nonlinear Science and Numerical Simulation, vol.48, pp.150-169, 2017.

D. Cintra, P. Argoul, ;. Rega, and F. Vestroni, Non-linear argumental oscillators: Stability criterion and approximate implicit analytic solution, International Journal of Non-Linear Mechanics, vol.94, pp.109-124, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01701277

D. Doubochinski, Argumental oscillations. Macroscopic quantum effects. SciTech Library, 2015.

D. I. Penner, D. B. Duboshinskii, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Phys. Usp, vol.16, issue.1, pp.158-160, 1973.

B. Pratiher and S. Kumar-dwivedy, Nonlinear response of a flexible cartesian manipulator withpayload and pulsating axial force, Nonlinear Dynamics, vol.57, issue.1, pp.177-195, 2009.

J. P. Treilhou, J. Coutelier, J. J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers, Advances in Space Research, vol.26, issue.9, pp.1423-1426, 2000.

D. Cintra and P. Argoul, Attractors capture probability in nonlinear argumental oscillators, Communications in Nonlinear Science and Numerical Simulation, vol.48, pp.150-169, 2017.

D. Doubochinski, Argumental oscillations. Macroscopic quantum effects, SciTech Library, 2015.

D. I. Penner, D. B. Duboshinskii, M. I. Kozakov, A. S. Vermel, and Y. V. Galkin, Asynchronous excitation of undamped oscillations, Phys. Usp, vol.16, issue.1, pp.158-160, 1973.

B. Pratiher and S. K. Dwivedy, Nonlinear response of a flexible cartesian manipulator withpayload and pulsating axial force, Nonlinear Dynamics, vol.57, issue.1, pp.177-195, 2009.

J. Treilhou, J. Coutelier, J. Thocaven, and C. Jacquez, Payload motions detected by balloon-borne fluxgate-type magnetometers, Advances in Space Research, vol.26, issue.9, pp.1423-1426, 2000.