Frequency Stabilization in a MEMS Oscillator with 1:2 Internal Resonance

A nonlinear piezoelectric shunt absorber with a 2:1 internal resonance: Theory

“…This is called a 𝑝:𝑞 internal resonance and (𝜔 𝑖 , 𝜔 𝑗 ) can be the (linear) natural frequencies of the modes as well as there nonlinear extension (the frequencies of the nonlinear modes, that depend on the amplitude of the motion). On the first case, the internal resonance is observed at low amplitude and is often a consequence of a particular geometry, obtained with symmetries (1:1 internal resonance are encountered in beams/strings of symmetric cross section [31], in circular/square plates [32,33], in cylindrical shells [34] and spherical caps [35]) or by intentional tuning, in musical instruments (1:2, 1:2:4 and 1:2:2:4:4:8 internal resonances are encountered in gongs and steel-pans [36,37]) or in micro-systems applications [38][39][40]. In the second case of an internal resonance with the nonlinear free oscillation frequencies, the coupling appear at larger amplitude, when the change of frequencies due to the amplitude of the motion is compatible with the frequency relationship (see e.g.…”

One-to-two internal resonance in a micro-mechanical resonator with strong Duffing nonlinearity

“…As a result, there has been significant interest in suppressing the nonlinearity of MEMS resonators to enhance their frequency stability [15][16][17][18][19]. Recently, researchers have proposed utilizing the non-linear phenomenon of internal resonance (InRes) as a new approach to stabilizing frequency fluctuations [20][21][22][23][24][25][26][27][28]. In a non-linear system, InRes can arise in an undriven vibrational mode by internally transferring energy from another vibrational mode that is externally driven when the two modal frequencies are commensurable into an m:n ratio (where m and n are integers) [29,30].…”

“…In recent years, advancements in InRes research have led to significant progress in understanding the underlying mechanisms of frequency stabilization. The pioneering experimental study by Antonio et al [22] demonstrated the use of 1:3 InRes to reduce frequency fluctuation, sparking further exploration in theoretical and experimental studies exploring 1:2 and 1:3 InRes for frequency stabilization [20][21][22][25][26][27][28]. Even though the direct comparison between different MEMS designs is not always feasible, it appears that 1:3 InRes can achieve better stabilization than 1:2 InRes.…”

A Robust Angular Rate Sensor Utilizing 2:1 Auto-Parametric Resonance Excitation

“…Due to modal interactions through AR, the nonlinearities and commensurable linear resonant frequencies in MEMS resonators may cause the resonator response in corresponding modes to be strongly coupled and dramatically amplified yet bounded by the damping and nonlinearities present in the system. Different cases of AR, including 1:1, 1:2, 2:1, 1:3, and 3:1 based on electrostatic actuation have been demonstrated in various studies of MEMS resonators [ 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ]. We present some of these relevant AR studies in MEMS resonators and the challenges to successfully implementing AR.…”

“…On the other hand, studies in [ 25 , 27 , 28 , 29 ] demonstrated how AR could be utilized in designing self-sustaining MEMS oscillators with nonlinearly coupled modes to output the desired reference frequency with fewer fluctuations. In [ 25 ], a fixed-fixed microbeam capable of resonating at 2:1 AR was experimentally tested using a closed loop system, and frequency stability of 13 ppm was achieved for the designed MEMS oscillator. These studies indicate AR-based MEMS resonators’ potential to achieve very low-frequency noise performance while operating them in the nonlinear regime.…”