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The present nonlinear study is motivated by the measured multiple simultaneous resonant frequency peaks of a MEMS resonator [1]. Above a sufficient power of the excitation signal, the response contains multiple resonant peaks. For a fixed power of the excitation, the peak with the lowermost frequency has a hardening bending that builds up from the first in-plane frequency F_1 and have its hysteresis jump down at F_ir. At the same time, there are peaks at 2*F_ir and 3*F_ir and if the power is sufficiently high, they are also bent with a hysteresis jumps down at 2*F_ir and 3*F_ir. Such a simultaneous frequency response with high corresponding amplitudes at the multiples of F_ir had been unexpected. Thus, it arose a demand for a model that can clarify it.

In this work, we propose a single degree of freedom mathematical model that can explain the nonlinear effect of frequency mixing. The resonant peaks of the model will be examined near F_1 , 2*F_1 and 3*F_1 by time series simulations and numerical continuation. The simultaneous resonance at different frequencies is expected due to the quadratic and cubic terms which are allowed in the governing ODE [2]. Briefly, such a nonlinear dynamics due to single vibrational mode under capacitive harmonic forcing in a MEMS beam resonator can be presented by an inhomogeneous second order ODE [3,4].
This ODE can describe the bended peak of hardening type [3] and the existence of simultaneous response around 2*F_1 and
3*F_1 [2,5,6]. When substituted in the quadratic and the cubic terms, the harmonic solution is treated with trigonometric identities which result in new harmonic terms that contain two and three times the harmonic frequency of the forcing. Also, the forcing has harmonic terms that correspond to 2*F_1 , too. It has to be noted that 2nd and 3rd, in- and out-of-plane harmonics of the micro-beam, have different values than 2*F_ir and 3*F_ir . Hence, the hysteresis jumps down at 2*F_ir and 3*F_ir must be secondary subharmonic resonances of the proposed single degree of freedom system excited near F_1

We aim at practical model that can explain anharmonic oscillations in NEMS devices. If successful, it will help the design of MEMS devices by predicting if the multiplied frequencies of the shifted first harmonics are going to appear in the response. As it is measured, this is an important feature due to the possible high amplitudes at 2*F_ir and 3*F_ir [1]. Also, when multiple resonant peaks are present, the hardening behavior is suitable for practical applications due to a dBm-range at which the hysteresis jump down appears at the specific value f_ir [1].

[1] F. Torres, G. Vidal-lvarez, A. Uranga, N. Barniol, Enhancement of Higher Harmonics Detectability in a Nonlinear Nanores-
onator, Proceedings of Symposium on Design, Test, Integration and Packaging of MEMS and MOEMS, April, 2014.
[2] A. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley, New York, 2007.
[3] Y. Yang, E. Ng, P. Polunin, Y. Chen, I. Flader, S. Shaw, M. Dykman, T. Kenny, Nonlinearity of Degenerately Doped
Bulk-Mode Silicon MEMS Resonators, Journal of Microelectromechanical Systems, 25, 5, 859-869, 2016.
[4] J. Lopez Mendez, Application of CMOS-MEMS Integrated Resonators to RF Communication Systems, PhD Thesis, Uni-
versitat Autonoma de Barcelona, 2009.
[5] L. Landau, E. Lifshitz, Mechanics, Pergamon Press, Oxford, Second edition, 1969.
[6] L. Landau, E. Lifshitz, Theory of elasticity, Pergamon Press, Oxford, Second edition, 1970.