Enhancing sensitivity in mode-localized tilt sensors based on asymmetrically coupled resonators

Mode-localized tilt sensors with amplitude ratio (AR) outputs have been proven to have a significant improvement in sensitivity over conventional frequency outputs, and the coupling stiffness between the coupled resonators is the key to sensitivity. The resonators in tilt sensors are designed identically to form coupling and have an equivalent response to the external perturbation. This paper proposes a new method to improve the sensitivity without weakening the coupling stiffness between resonators by exploiting the asymmetry of coupled resonators. Using the amplitude ratio of the asymmetrically resonators with a smaller effective mass to the resonator with a larger effective mass as the output, the sensitivity of the mode-localized tilt sensor will be increased approximately by a factor of the ratio of the effective masses of the two resonators compared to symmetrically coupled resonators. Sensitivity can be increased to twice the original using asymmetrically coupled resonators, which has been verified both in theoretical analysis and finite element method (FEM) simulation.


Introduction
Due to the advantages of small size, light weight, low cost, high precision, and low power consumption, MEMS tilt sensors are widely used in satellite autonomous positioning, aerospace, consumer electronics, precision equipment calibration, bridge erection [1][2][3], etc.According to the different sensing mechanisms, MEMS tilt sensors can be divided into resonant sensing [4], capacitive sensing [5], impedance sensing [6], mode-localized sensing [7][8], etc.The mode localization effect has already been widely used in acceleration sensing [9][10][11], tilt sensing [7][8], mass sensing [12][13], and charge sensing [14][15] in recent years.Compared with the traditional frequency output, the amplitude ratio output has a significant improvement in sensitivity and shows common mode rejection to the environment [16].The mode-Localized MEMS tilt sensor was reported in [7] for the first time by using two mode-localized resonant accelerometers.
The sensitivity is one of the most important parameters of the mode-localized tilt sensors and the coupling stiffness between the coupled resonators is the key to sensitivity.A higher degree of freedom resonator system or resonators working in the high order mode [13,17] has been proven can also enhance the sensitivity, while the size of the device will be larger.The two resonators in the mode-localized MEMS tilt sensor are identically designed with the same parameters and have an equivalent response to the external perturbation, coupling by the electrostatic structure or the mechanical structure.Two specially designed cantilevers with different lengths have been used in the modelocalized mass sensor [12], which demonstrates that resonators with the same resonant frequency but different sizes can also be coupled.A well-designed asymmetry has been proven to enhance the sensitivity of sensing applications that rely on the mode localization effect [18], but only theoretical analysis and numerical simulations have been performed.
This article presents a novel approach to enhance sensitivity without weakening the coupling stiffness by exploiting asymmetrically coupled resonators rather than identical ones.The dynamic equations of motion of the asymmetrically coupled resonators are revised, and the theoretical analysis proves that the sensitivity of the mode-localized tilt sensor will be increased approximately by a factor of the ratio of the effective masses of the two resonators compared to symmetrically coupled resonators.The improvement in sensitivity using asymmetrically coupled resonators is verified in FEM simulation and is consistent with the theoretical analysis.

Working principle of the resonant mode-localized tilt sensor
The schematic view of the mode-localized tilt sensor with asymmetrically coupled resonators is shown in Figure 1.The component of the acceleration of gravity ( g ) due to the shift in the angle acts on the proof mass to generate inertial stress.The inertial stress works on the weakly asymmetrically coupled resonators after being amplified by a one-stage micro-lever structure, causing the energy to be redistributed between the asymmetrically coupled resonators, resulting in the mode localization effect.The ratio of vibration amplitude between the asymmetrically coupled resonators is available for the output as it is a function of the external angle perturbations.The asymmetry of the coupled resonators is that the resonators with different effective masses have the same resonant frequency, and the resonator (resonator 1) connected to the micro-lever structure has a smaller effective mass compared to the other resonator (resonator 2).The sensitivity of the mode-localized tilt sensor will be improved by using the amplitude ratio of resonator 1 to resonator 2 compared to traditional symmetrically coupled resonators.

Dynamics equations of asymmetrically coupled resonators
The weakly asymmetrically coupled resonators can be viewed as a 2-DoF mass-spring-damper system.The effective mass and intrinsic stiffness of the two resonators are assumed to have a certain proportional relationship, i.e.
represents proof mass, k  represents the effective scale factor of stress to stiffness).Assuming the stiffness perturbation is applied to resonator 1, i.e. 1 k k k    .The dynamic equations of motions of the 2-DoF weakly asymmetrically coupled resonators system can be described as [19]     where x , x  , x  , f and c , respectively, represent the displacement of the resonator, velocity of the resonator, acceleration of the resonator, single-sided driving of the resonator, and damping coefficient.After Laplace transform, the equations can be expressed as The resonant frequencies of the weakly asymmetrically coupled modes can be expressed as   and   2 0 F j  , the expression of the AR of the two weakly asymmetrically coupled resonators under out-of-phase mode can be expressed as The sensitivity of AR can be obtained by

Numerical simulation
From the above AR sensitivity formula, it can be concluded that the asymmetric effective mass of the coupled resonator can indeed enhance the sensitivity by a scale factor of the ratio of the effective masses.The numerical simulation is established in Matlab according to the theoretical model and the specific values of the key parameter are shown in Table 1.When the effective mass ratio of two resonators is greater than or equal to 4, it is difficult to ensure that they have the same resonant frequency in device design even with electrostatic force tuning.Therefore, the case where N is greater than or equal to 4 is ignored.The mode resonant frequency shift curve with different angles applied to the tilt sensor under different N is shown in Figure 2. The minimum frequency difference between the two coupled modes will be reduced with the increase of N, as shown in Figure 3, which is similar to the principle of weakening the coupling stiffness between coupled resonators.The amplitude ratio shift curve with different angles applied to the tilt sensor under different N is shown in Figure 4.The sensitivity in the linear operation range can be improved by a scale factor of N (ratio of the effective masses of the two resonators) compared to symmetrically coupled resonators, as shown in Figure 5.

FEM simulation verification
Figure 6 displays the FEM simulation model of the mode-localized tilt sensor using COMSOL 5.6 and the weakly asymmetrically coupled modes of asymmetrically coupled resonators at different angles.The mechanical structure of the mode-localized tilt sensor composed of Si material layer is placed in an air box (grey area in Figure 6(a)) to generate electrostatic force.The anchors (yellow area in Figure 6(a)) of the tilt sensor are set fixed, and the other part (red area of Figure 6(a)) of the tilt sensor is set free.The mesh is divided into free tetrahedral meshes.The effect of angular shifts can be simulated by applying a body force in the direction of the sensitive axis to the proof mass in the physical field of solid mechanics.The resonant frequencies of the asymmetrically coupled resonators and the sensitivity of the tilt sensor using AR as the output are simulated through the FEM simulation.2. The effective mass of resonator 2 is increased to about 1.93 times of resonator 1 in the FEM simulation.FEM simulation results of the mode resonant frequency shift curve with the angle varying from -30° to 30° are shown in Figure 7, which indicates a mode localization effect between the asymmetric resonators.The sensitivity of AR in the linear operation range can be improved to 2.10 times from 0.392AR/°to 0.824AR/°as shown in Figure 8.The improvement of sensitivity is almost the ratio of the effective mass of the two resonators, which is consistent with the theoretical analysis results.

Conclusions
In this paper, a new approach is presented to enhance the sensitivity in Mode-localized tilt sensors by exploiting the asymmetry of coupled resonators.The resonator with a small mass is used to sense shift in external angles and the ratio of the vibration amplitude of the resonator with a smaller effective mass to the resonator with a larger effective mass as the output.The sensitivity is improved by approximately the ratio of the effective masses of the two resonators, which has been verified in theoretical analysis and FEM simulation.Two coupled resonators with different effective masses are guaranteed to have the same resonant frequency, which will propose a great challenge to the fabrication process.The Mode-localized tilt sensor based on asymmetrically coupled resonators will be fabricated and tested in the future.

Figure 1 .
Figure 1.The schematic diagram of mode-localized tilt sensor with asymmetric coupled resonators.
1 2 / m m N m   and 1 2 / k k N k   (N is the ratio of the effective mass of resonator 2 to resonator 1), respectively, representing the effective mass and intrinsic stiffness of the weakly coupled resonators.c k represents the effective coupling stiffness with c k k  and k  represents the effective stiffness perturbation due to the shift in angle ( sin( )

Figure 2 .
Figure 2. Mode resonant frequency shift curve with the angle range from -90° to 90° under different N.

Figure 3 .
Figure 3. Mode resonant frequency shift curve with the angle varying from -10° to 10° under different N.

Figure 4 .
Figure 4. AR shift curve with the angle range from -90° to 90° under different N.

Figure 5 .
Figure 5. AR shift curve with the angle varying from -30° to 30° under different N.

Figure 6 .
Figure 6.(a) FEM simulation model of the mode-localized tilt sensor, (b) Vibration modes of asymmetrically coupled resonators at different angles.Resonator 1 and resonator 2 are designed as clamped-clamped tine beams and weakly coupled by a mechanical coupling structure.The effective mass of resonator 2 can be increased by enlarging the length and width in a certain proportion and the resonant frequency of resonator 2 remains the same as resonator 1.The key parameters of the mode-localized tilt sensor are shown in Table2.The effective mass of resonator 2 is increased to about 1.93 times of resonator 1 in the FEM simulation.FEM simulation results of the mode resonant frequency shift curve with the angle varying from -30° to 30° are shown in Figure7, which indicates a mode localization effect between the asymmetric resonators.The sensitivity of AR in the linear operation range can be improved to 2.10 times from 0.392AR/°to 0.824AR/°as shown in Figure8.The improvement of sensitivity is almost the ratio of the effective mass of the two resonators, which is consistent with the theoretical analysis results.

Figure 7 .
Figure 7.The mode resonant frequency shift curve with the angle varying from -30° to 30° through FEM simulation.

Figure 8 .
Figure 8. AR shift curve with the angle varying from -30° to 30° through FEM simulation.

Table 1 .
Numerical simulation parameters of the mode-localized tilt sensor.

Table 2 .
The key parameters of the mode-localized tilt sensor.