Mechanically coupled MEMS resonant accelerometer based on internal resonance and detection method thereof

Abstract

The application belongs to the technical field of MEMS resonant sensor, and particularly discloses a mechanical coupling MEMS resonant accelerometer based on internal resonance and a detection method thereof. The accelerometer comprises a movable mass block and force amplification levers arranged at four corners of the movable mass block. Two pairs of mechanical coupling resonant beams are symmetrically arranged at two sides of the movable mass block. A plurality of electrodes are arranged at two sides of each set of the mechanical coupling resonant beams. The mechanical coupling resonant beam comprises an external resonant beam and an internal resonant beam. The external resonant beam and the internal resonant beam are connected through a mechanical coupling beam. The mechanical coupling MEMS resonant accelerometer based on internal resonance and the detection method thereof can effectively inhibit the detection error caused by alternating current driving voltage disturbance, greatly reduce the resonant frequency Allan deviation, significantly improve the frequency stability, detection robustness and weak acceleration detection capability, and are suitable for high-precision inertial navigation, precise vibration detection and other fields.

Description

Technical Field

[0001] This invention relates to the field of MEMS (Micro-Electro-Mechanical Systems) resonant sensor technology, and in particular to a mechanically coupled MEMS resonant accelerometer based on internal resonance and its detection method. Background Technology

[0002] MEMS (Micro-Electro-Mechanical Systems) resonant accelerometers identify changes in external acceleration by measuring changes in their resonant frequency. They offer advantages such as low power consumption, miniaturization, and ease of integration, making them promising for applications in smart wearable devices, communications, and inertial navigation. In practical applications, the stability of the resonant frequency is crucial to the accelerometer's resolution and reliability; fluctuations in the resonant frequency can mask weak measurement signals and affect detection performance.

[0003] However, due to nonlinear driving forces and geometric nonlinearity, resonant devices inevitably operate within a nonlinear vibration range. The dependence between peak frequency and amplitude further leads to a decrease in the stability of the accelerometer's resonant frequency. Through energy transfer between different resonant units, the mechanical frequency lock (MFL) phenomenon can be excited, significantly suppressing frequency fluctuations in the nonlinear resonant system, thereby improving the frequency stability and performance of the resonant device.

[0004] When energy transfer occurs between different vibration modes due to internal resonance, the peak frequency of a nonlinear resonant system no longer depends on the amplitude of the driving force; this is known as the mechanical frequency-locked (MFL) phenomenon. Once MFL occurs, frequency fluctuations caused by stiffness softening and hardening in the nonlinear resonant system can be suppressed during both forward and reverse frequency sweeps. For example, Antonio et al. experimentally measured energy exchange between different vibration modes and implemented MFL using a stable mechanical negative feedback loop. Xu et al. proposed a novel frequency stabilization scheme utilizing the inherent high-order nonlinear interactions in a parametrically driven arched beam resonator. Similarly, Zhang et al. successfully demonstrated a frequency reduction of Allen bias through internal resonance between tension and bending modes in a U-shaped resonator. Yu et al. implemented MFL and improved frequency stability using a 1:2 internal resonance mechanism in a high-Q sealed beam resonator. Considering that MFL can provide a stable peak frequency, preliminary sensing methods based on MFL have been reported. For example, Li et al. implemented a novel trigger-based acceleration warning method by switching between frequency-locked and unlocked states. However, the application of MFL in quantitative detection by resonant accelerometers still requires further research.

[0005] Sensitivity and resolution are crucial performance indicators for MEMS resonant accelerometers. Traditional resonant sensing methods primarily utilize changes in linear resonant frequencies, resulting in limited sensitivity. Due to geometric and driving nonlinearities, resonant devices operate within a nonlinear vibration range, leading to limited resolution. Various methods have been explored to improve the sensitivity and resolution of resonant accelerometers. For example, Ma et al. designed a novel MEMS resonant accelerometer based on mechanical stiffness coordination and an active damping system. Their research found that the active damping system helps the prototype protect against external vibration noise, thereby improving bias stability. Kose et al. proposed a high-performance MEMS temperature sensor based on a double-ended tuning fork resonator and a strain amplification beam, increasing the resonator's frequency temperature coefficient by 33 times. Typically, micro-lever mechanisms are also introduced to enhance sensitivity by amplifying the inertial force induced by the acceleration signal.

[0006] Furthermore, nonlinear vibrations hold great potential for improving sensor performance. For example, Wang et al. reported a weakly coupled double-ended tuning fork resonator under strong parametric modulation. Their research found that the nonlinear parametrically driven sensor can exhibit a two-order-of-magnitude improvement in sensitivity while maintaining a wide measurement range. Wu et al. proposed a resonance detection method based on the bifurcation phenomenon caused by modal coupling vibration. Compared with traditional frequency shift detection methods, this bifurcation-based detection method improves sensitivity by 5.6 times. Introducing the mechanical frequency-locked (MFL) phenomenon into resonant accelerometers holds promise for achieving new breakthroughs in performance metrics.

[0007] In summary, nonlinear vibrations demonstrate significant application potential in resonant sensing. The mechanical frequency locking (MFL) phenomenon caused by internal resonance can enhance the frequency stability of nonlinear resonant systems, which is of great importance for improving the performance of resonant accelerometers. Summary of the Invention

[0008] The purpose of this invention is to provide a mechanically coupled MEMS resonant accelerometer based on internal resonance and its detection method, so as to solve the problems of limited sensitivity, poor frequency stability and insufficient robustness of existing resonant accelerometers. By using 1:1 internal resonance to excite the mechanical frequency locking (MFL) phenomenon, the dependence effect of response frequency and amplitude is overcome, and the detection resolution and robustness are improved.

[0009] To achieve the above objectives, the present invention provides a mechanically coupled MEMS resonant accelerometer based on internal resonance, comprising a movable mass block and force amplification levers disposed at the four corners of the movable mass block. Two pairs of mechanically coupled resonant beams are symmetrically arranged on both sides of the movable mass block. Several electrodes are disposed on both sides of each pair of mechanically coupled resonant beams. The mechanically coupled resonant beams include an external resonant beam and an internal resonant beam, and the external resonant beam and the internal resonant beam are connected by the mechanically coupled beam.

[0010] Preferably, the movable mass block is used to receive external acceleration signals and generate inertial force; The force amplification lever is used to amplify the inertial force generated by the movable mass block and transmit it to the internal resonant beam. The electrodes include a driving electrode, a detection electrode, and a stiffness adjustment electrode; the driving electrode is used to apply a swept-frequency AC driving signal; the detection electrode is used to acquire and output the vibration signal of the resonant beam; the stiffness adjustment electrode is used to apply a DC voltage to adjust the fundamental frequency of the internal resonant beam.

[0011] Preferably, the driving electrode and the detection electrode are provided on both the left and right sides of the external resonant beam; An AC driving voltage is applied to the driving electrode. When the AC driving voltage exceeds 180mV, energy is transferred between the external resonant beam and the internal resonant beam through a 1:1 internal resonance, triggering the mechanical frequency lock (MFL) phenomenon.

[0012] Preferably, the quality factor of the external resonant beam is 8000, and the quality factor of the internal resonant beam is 34000.

[0013] Preferably, the stiffness adjustment electrodes are symmetrically distributed on the upper and lower sides of the internal resonant beam. A DC voltage in the range of 0V to 70V is applied to the stiffness adjustment electrodes to compensate for the fundamental frequency difference between the external resonant beam and the internal resonant beam caused by processing errors and residual stress, so that the external resonant beam and the internal resonant beam generate a 1:1 internal resonance and excite the mechanical frequency lock-in (MFL) phenomenon.

[0014] Preferably, both the movable mass block and the mechanically coupled resonant beam are grounded.

[0015] This invention also provides a detection method for a mechanically coupled MEMS resonant accelerometer based on internal resonance, comprising the following steps: S1. Apply a DC driving voltage to the internal resonant beam and adjust the fundamental frequency of the internal resonant beam so that the fundamental frequency of the external resonant beam is close to that of the internal resonant beam, so as to meet the triggering condition of 1:1 internal resonance. S2. Apply an AC drive signal to the drive electrode of the external resonant beam. When the amplitude of the AC drive signal is greater than or equal to the critical value of the mechanical frequency lock-in MFL phenomenon, the 1:1 internal resonance is excited to generate the mechanical frequency lock-in MFL phenomenon. S3. Apply different acceleration signals to the movable mass block, measure the peak frequency of the external resonant beam, and determine the acceleration signal to be measured based on the change in peak frequency. S4. By testing the Allen deviation of the resonant frequency, evaluate the minimum resolution of the accelerometer under different resonant conditions.

[0016] Preferably, S1 includes the following steps: S11. Apply a DC voltage ranging from 0V to 70V between the symmetrical stiffness adjustment electrodes on the upper and lower sides of the internal resonant beam. A lock-in amplifier is used to generate a swept-frequency AC signal, which is applied to the upper electrode. The lower electrode outputs a corresponding induced current signal, which is amplified by a cross-group amplifier and observed on the lock-in amplifier. S12. Apply a pair of DC voltages to the driving electrodes on the left and right sides of the external resonant beam. A swept-frequency AC signal is applied to the left electrode of the external resonant beam, and the corresponding induced current signal is output from the right electrode. After being amplified by the cross-group amplifier, it is observed on the lock-in amplifier.

[0017] Preferably, in S2, the critical value of the mechanical frequency lock-in (MFL) phenomenon is the parameter value corresponding to the merging of the vibration peaks of the external and internal resonant beams in the amplitude-frequency response curve of the accelerometer into one, and the peak frequency locks at 174.5kHz. When the mechanical frequency lock-in (MFL) phenomenon occurs, the following dynamic equations of the mechanically coupled resonant system are satisfied: ; ; in, and These represent the amplitudes of the external and internal resonant beams, respectively. and These represent the instantaneous velocities of the external and internal resonant beams, respectively. and These represent the instantaneous accelerations of the external and internal resonant beams, respectively. and These represent the fundamental frequencies of the external and internal resonant beams, respectively. and These represent the damping of the external resonant beam and the internal resonant beam, respectively. This represents the mechanical coupling stiffness between the external and internal resonant beams. and Indicates the nonlinear stiffness coefficient; and This indicates the amplitude and frequency of the electrostatic driving force.

[0018] Preferably, S4 includes the following steps: S41. Perform static testing: Place the MEMS accelerometer horizontally and record the resonant frequency for 5 minutes. The sampling frequency is 1600 Sa / s. Measure the minimum Allen deviation under different resonant conditions. S42. Based on the proportional relationship between resolution and Allen deviation, combined with the resolution formula of a single-sided resonant structure, and considering the symmetry of the resonant system, the minimum resolution of the differential resonant accelerometer is predicted by measuring the sensitivity and Allen variance of the single-sided resonant structure. S43. Substitute Allen bias and sensitivity into the formula to obtain the resolution under different resonance conditions.

[0019] Therefore, the present invention employs the above-mentioned mechanically coupled MEMS resonant accelerometer and its detection method based on internal resonance, and the beneficial effects are as follows: (1) Based on the mechanical frequency locking (MFL) phenomenon of 1:1 internal resonance excitation, this invention can effectively suppress the Allen deviation of the resonant frequency and reduce the minimum resolution of the accelerometer to 3.7µg, which is far superior to the traditional resonant detection method and greatly enhances the detection capability of weak acceleration signals.

[0020] (2) The mechanical frequency locking MFL phenomenon of the present invention can overcome the response frequency and amplitude dependence effect caused by stiffness hardening in nonlinear vibration, suppress the detection error caused by AC drive voltage fluctuation, avoid the influence of driving force disturbance on peak frequency in closed-loop experiment, and improve the reliability of device in practical application.

[0021] (3) By adjusting the DC drive voltage of the internal resonant beam, the accelerometer sensitivity can be increased from 312.6 Hz / g to 366 Hz / g. At the same time, the mechanical frequency lock (MFL) phenomenon can ensure frequency stability without significantly affecting the sensitivity, thus achieving synergistic optimization of sensitivity and stability.

[0022] The technical solution of the present invention will be further described in detail below with reference to the accompanying drawings and embodiments. Attached Figure Description

[0023] Figure 1 This is a schematic diagram of the overall structure of the mechanically coupled MEMS resonant accelerometer embodiment based on internal resonance of the present invention. Among them, (a) is a structural design diagram of the MEMS resonant accelerometer, (b) is a schematic diagram of the force amplification lever, (c) is a schematic diagram of the mechanically coupled resonance structure, (d) is a schematic diagram of the MEMS resonant accelerometer product and an electron microscope image, (e) is an enlarged schematic diagram of the letter X in (d), and (f) is an enlarged schematic diagram of the letter Y in (e). Figure 2This invention relates to a manufacturing process flow for a MEMS resonant accelerometer based on an internal resonance mechanically coupled MEMS resonant accelerometer embodiment. Step (a) involves photolithography etching on a bottom surface coated with photoresist, using a first mask; step (b) involves dry etching of the back cavity layer region; step (c) involves buffered oxidation etching of the oxide insulating layer; step (d) involves wet etching of the electrode pattern; step (e) involves covering the upper surface of the device with photoresist; and step (f) involves dry etching of the device layer, connecting it to the oxide insulating layer. Figure 3 This is an experimental measurement schematic diagram of the MEMS accelerometer according to an embodiment of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention, wherein (a) is the design of the accelerometer driving and detection circuit; and (b) is a schematic diagram of the experimental measurement device. Figure 4 This is a schematic diagram of the inherent frequency characteristics of the mechanically coupled resonant beam under DC voltage regulation and the amplitude-frequency of the first two modes at 65V, according to an embodiment of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention. (a) shows the effect of DC voltage on the first two inherent frequencies of the mechanically coupled resonant beam; (b) shows the amplitude-frequency of the first two modes of the mechanically coupled resonant beam considering a DC voltage of 65V; the first-order mode frequency on the left corresponds to X in (a), and the second-order mode frequency on the right corresponds to Y in (a). Figure 5 This is an embodiment of the detection method of mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention, considering the 15V DC voltage condition, and the amplitude-frequency response curves obtained by experiment, wherein (a) is the amplitude-frequency response curve of the external resonant beam and (b) is the amplitude-frequency response curve of the internal resonant beam. Figure 6 This invention relates to a detection method for a mechanically coupled MEMS resonant accelerometer based on internal resonance, considering a 65V DC voltage condition. The experimentally measured mechanical frequency lock (MFL) phenomenon and the theoretically predicted amplitude-frequency response curves are shown. (a) and (b) are the experimentally measured amplitude-frequency curves of the external and internal resonant beams, respectively, when the external resonant beam is driven. (c) and (d) are the theoretically predicted amplitude-frequency curves of the external and internal resonant beams, respectively, when the external resonant beam is driven. Figure 7 This is a schematic diagram of applying an acceleration signal to a movable mass block according to an embodiment of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention. In this diagram, (a) shows the relationship between acceleration and rotation angle; (b) shows a PCB board containing a MEMS accelerometer sensor, a driving circuit and a detection circuit, which is an enlarged schematic diagram of Z in (d); (c)-(e) show several typical configuration schemes for acceleration values. Figure 8The present invention provides experimental measurement results of the amplitude-frequency response curves of the internal resonant beam under different acceleration signals in an embodiment of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance, wherein (a) considers a DC voltage of 15V; and (b) considers a DC voltage of 65V. Figure 9 This invention relates to a detection method for a mechanically coupled MEMS resonant accelerometer based on internal resonance. When the mechanical frequency lock-up (MFL) phenomenon occurs, the amplitude-frequency response curves of the external resonant beam under different acceleration signals are experimentally measured. In this embodiment, (a) the AC driving voltage is considered to be 200mV; and (b) the AC driving voltage is considered to be 300mV. Figure 10 This invention relates to a detection method for a mechanically coupled MEMS resonant accelerometer based on internal resonance. The method involves real-time detection of the peak frequency change of the acceleration signal using a phase-locked loop. In this embodiment, (a) a DC voltage of 15V is considered; (b) no mechanical frequency lock-up (MFL) phenomenon occurs when the DC voltage is 65V; and (c) a mechanical frequency lock-up (MFL) phenomenon occurs when the DC voltage is 65V. Figure 11 The Allen variances of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention are experimentally measured under different resonance conditions. Among them, (a) is the Allen variance measured by driving the internal resonant beam under a DC voltage of 15V; (b) is the Allen variance measured by driving the internal resonant beam under a DC voltage of 65V; (c) is the Allen variance measured by driving the resonant beam externally under a DC voltage of 65V, at which time the mechanical frequency lock-in (MFL) phenomenon will occur. Figure 12 The figures show the resolution variation curves of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance according to the present invention, wherein (a) is the resolution variation curve of the internal resonant beam driven and detected under DC voltage conditions of 15V and 65V respectively, and (b) is the resolution variation curve of the external resonant beam driven and detected under DC voltage conditions of 65V. Figure 13 This is a coordinate diagram of the coupled resonant beam in an embodiment of the detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention. Detailed Implementation

[0024] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0025] Unless otherwise defined, the technical or scientific terms used in this invention shall have the ordinary meaning understood by one of ordinary skill in the art to which this invention pertains. The terms "first," "second," and similar terms used in this invention do not indicate any order, quantity, or importance, but are merely used to distinguish different components. Terms such as "comprising" or "including" mean that the element or object preceding the word encompasses the elements or objects listed following the word and their equivalents, without excluding other elements or objects.

[0026] like Figure 1 As shown in (a), the design model of the mechanically coupled MEMS resonant accelerometer based on internal resonance of the present invention includes a movable mass block and four force amplification levers disposed at the four corners of the movable mass block. The movable mass block is used to receive external acceleration signals and generate inertial forces. The force amplification levers are used to amplify the inertial forces generated by the movable mass block and transmit them to the internal resonant beams. Two pairs of mechanically coupled resonant beams are symmetrically arranged on both sides of the movable mass block.

[0027] The mechanically coupled resonant beam includes an external resonant beam and an internal resonant beam, which are connected by a mechanical coupling beam. Acceleration signals can be measured by detecting frequency changes in the resonant beams. To achieve driving, detection, and stiffness adjustment of the resonant beams, this invention provides several electrodes on both sides of each set of mechanically coupled resonant beams. These electrodes include driving electrodes, detection electrodes, and stiffness adjustment electrodes. The driving electrodes apply a swept-frequency AC driving signal; the detection electrodes acquire and output the vibration signal of the resonant beam; and the stiffness adjustment electrodes apply a DC voltage to adjust the fundamental frequency of the internal resonant beam. Specifically, driving and detection electrodes are provided on both the left and right sides of the external resonant beam, while the stiffness adjustment electrodes are symmetrically distributed on the upper and lower sides of the internal resonant beam.

[0028] Under operating conditions, a DC polarization voltage is applied to the driving and sensing electrodes, and both the movable mass and the mechanically coupled resonant beam are grounded. Typically, different DC voltages applied to the resonant system control the bending stiffness of the resonant beam. When the mass receives acceleration parallel to the x-direction of the resonant beam, the resulting inertial force acts on the inner resonant beam through an amplifying lever, thus affecting its natural frequency. The differential push-pull output response can then be obtained by measuring the change in the natural frequency of the resonant beam. Table 1 shows the structural design parameters of the MEMS resonant accelerometer. Table 1 Structural design parameters of MEMS resonant accelerometer

[0029] Finite element simulation showed that a lever mechanism consisting of four force-amplifying levers can amplify the inertial force of the mass block by approximately 13 times. Figure 1As shown in (b) of the diagram. To achieve the mechanical frequency locking (MFL) phenomenon caused by the 1:1 internal resonance, this invention introduces a mechanically coupled beam to realize the vibrational energy exchange between the external resonant beam (resonant beam 1) and the internal resonant beam (resonant beam 2), as follows: Figure 1 As shown in (c) above. However, due to processing errors and residual stress, there is a slight difference in the fundamental frequency of the two resonant beams. To compensate for the frequency difference and promote the occurrence of 1:1 internal resonance, different DC voltages are introduced on the upper and lower sides of the inner resonant beam to adjust its fundamental frequency. Finally, the tested MEMS resonant accelerometer was fabricated using commercial silicon-on-insulator micromachining technology, as shown in (c). Figure 1 As shown in (d) in the diagram. The specific process flow is as follows: Figure 2 As shown, the material is an SOI wafer with a top layer thickness of 25 μm. Figure 2 As shown in (a), photoresist is coated on the lower surface of a commercial insulator and a first mask is photolithographically etched; as Figure 2 As shown in (b), dry etching is performed in the cavity layer region; as Figure 2 As shown in (c), the oxide insulating layer undergoes buffered oxide etching; as Figure 2 As shown in (d), a wet etching electrode pattern is formed; as Figure 2 As shown in (e), the upper surface is coated with photoresist; as Figure 2 As shown in (f), the device layer is dry etched to the oxide insulating layer.

[0030] This invention aims to experimentally measure and theoretically predict the amplitude-frequency response characteristics of MEMS accelerometers under different resonance conditions, with particular attention to the mechanical frequency lock-in (MFL) phenomenon caused by the 1:1 internal resonance between the external and internal resonant beams. Considering the left-right symmetry of the resonant accelerometer, this invention uses the mechanically coupled resonant structure on the right side for experimental measurements.

[0031] This invention also provides a detection method for a mechanically coupled MEMS resonant accelerometer based on internal resonance, comprising the following steps: S1. Apply a DC driving voltage to the internal resonant beam and adjust its fundamental frequency to make the fundamental frequency of the external resonant beam close to that of the internal resonant beam, so as to meet the triggering condition of 1:1 internal resonance. This includes the following steps: S11. First, the design and experimental conditions of the accelerometer drive and detection circuit are discussed in detail. Experimental measurements show that, due to manufacturing errors, the fundamental frequency of the internal resonant beam is slightly higher than that of the external resonant beam. Therefore, this invention utilizes symmetrical electrodes at the top and bottom of the internal resonant beam to lower its natural frequency. First, a pair of DC voltages ranging from 0V to 70V are applied between the symmetrical stiffness adjustment electrodes on the upper and lower sides of the internal resonant beam. Then, a lock-in amplifier is used to generate a swept-frequency AC signal, which is applied to the upper electrode; the lower electrode outputs a corresponding induced current signal, which is amplified by a cross-group amplifier and observed on the lock-in amplifier. Figure 3 As shown in (a) of the diagram.

[0032] S12. Apply a pair of DC voltages to the driving electrodes on the left and right sides of the external resonant beam. A swept-frequency AC signal is applied to the left electrode of the external resonant beam, and the corresponding induced current signal is output from the right electrode. This signal is amplified by a cross-group amplifier and then observed by a lock-in amplifier. The MEMS accelerometer of this invention is then placed in a vacuum cavity, and the experimental measurement scenario is as follows. Figure 3 As shown in (b) of the figure. By using half-power bandwidth analysis test data, the quality factors of the external resonant beam and the internal resonant beam are approximately 8000 and 34000, respectively.

[0033] Considering that when the DC drive voltage is very low, the first mode shape of the accelerometer mainly exhibits bending vibration of the external resonant beam. Similarly, the second mode shape mainly exhibits bending vibration of the internal resonant beam, and the first two natural frequencies of the accelerometer are close to 171kHz and 181kHz, respectively. Experimental measurements show that as the DC drive voltage... With the increase of , the first natural frequency did not change significantly, while the second natural frequency decreased significantly, such as Figure 4 As shown in (a) above. To compare the traditional resonant sensing method with the frequency-locked induced sensing method, this experiment mainly focuses on DC voltage. The accelerometer's operating performance is at 15V and 65V. This is because when the DC voltage... At 65V, the first two natural frequencies are relatively close, such as Figure 4 As shown in (b), this may trigger the mechanical frequency lock-up (MFL) phenomenon of the accelerometer caused by the 1:1 internal resonance.

[0034] S2. This step aims to study the amplitude-frequency response characteristics of the accelerometer under different resonance conditions. In particular, the mechanical frequency lock-in (MFL) phenomenon was experimentally measured and theoretically predicted. First, when the DC voltage... At 15V, the vibration behavior of the external and internal resonant beams was experimentally measured, as follows: Figure 5 As shown, the forward frequency sweep experiment exhibits significant stiffness hardening. Furthermore, due to the significant difference in the natural frequencies of the two resonant beams, it is difficult to observe internal resonance behavior and mechanical frequency locking (MFL) phenomenon.

[0035] Then, consider DC voltage. At 65V, the natural frequency difference between the two resonant beams decreases, which is beneficial for achieving 1:1 internal resonance behavior and mechanical frequency locking (MFL) phenomenon. Different AC drive signals were applied to the driving electrodes of the external resonant beam, and the amplitude-frequency response curves of the two resonant beams were experimentally measured.

[0036] By scanning the drive frequency upwards in the experiment, it was found that when the amplitude of the AC drive signal is greater than or equal to the critical value of the mechanical frequency lock-in (MFL) phenomenon, it will excite the 1:1 internal resonance and generate the mechanical frequency lock-in (MFL) phenomenon. The critical value of the mechanical frequency lock-in (MFL) phenomenon is the parameter value corresponding to the merging of the vibration peaks of the external resonant beam and the internal resonant beam in the amplitude-frequency response curve of the accelerometer, and the peak frequency locks at 174.5kHz. In this experiment, it is 180mV.

[0037] For AC drive voltages below 180mV, the peak frequency increases significantly with drive strength. However, when the AC drive voltage exceeds 180mV, due to energy transfer between different resonant beams via 1:1 internal resonance, the peak frequency of the resonant beam stabilizes at around 174.5kHz, leading to the mechanical frequency lock-in (MFL) phenomenon, such as... Figure 6 As shown in (a) of the diagram.

[0038] Furthermore, the mechanical frequency lock-in (MFL) phenomenon was experimentally observed by measuring the vibration of the internal resonant beam. It is important to note that when the AC drive voltage is below 180mV, the internal resonant beam exhibits two distinct vibration peaks, corresponding to the resonant frequencies of the two beams, as shown below. Figure 6 As shown in (b) of the diagram.

[0039] When the mechanical frequency lock-in (MFL) phenomenon occurs, the following dynamic equations of the mechanically coupled resonant system are satisfied: ; ; in, and These represent the amplitudes of the external and internal resonant beams, respectively. and These represent the instantaneous velocities of the external and internal resonant beams, respectively. and These represent the instantaneous accelerations of the external and internal resonant beams, respectively. and These represent the fundamental frequencies of the external and internal resonant beams, respectively. and These represent the damping of the external resonant beam and the internal resonant beam, respectively. This represents the mechanical coupling stiffness between the external and internal resonant beams. and Indicates the nonlinear stiffness coefficient; and This indicates the amplitude and frequency of the electrostatic driving force.

[0040] Considering < A 1:1 internal resonance behavior may occur between the two resonant beams. Based on previous research, it was found that due to 1:1 internal resonance and stiffness hardening, as the driving force of the low-frequency resonant beam increases, the response frequency of the resonant system will lock near the natural frequency of the high-frequency resonant beam, thus producing a mechanical frequency lock-in (MFL) phenomenon. Then, the mechanical frequency lock-in (MFL) phenomenon was theoretically predicted. By solving the dynamic equations of the mechanically coupled resonant system using the long-time integration method, the amplitude-frequency response curves of the external and internal resonant beams under different AC driving voltages were theoretically predicted. With the increase of the AC driving voltage, the peak frequencies of both the external and internal resonant beams are locked near the second-order natural frequency, which is qualitatively consistent with experimental measurements. Figure 6 As shown in (c)-(d). It is important to note that when the peak resonant frequency of the resonant system reaches the fundamental frequency of the internal resonant beam, mechanical frequency lock-in (MFL) occurs. Based on the structure of the MEMS accelerometer, the acceleration signal can influence the natural frequency of the internal resonant beam through the inertial force of the mass block. Therefore, the acceleration signal can be sensed by detecting changes in the lock-in frequency.

[0041] S3. Apply different acceleration signals to the movable mass block, measure the peak frequency of the external resonant beam, and determine the acceleration signal to be measured based on the change in peak frequency.

[0042] This step involved experimental measurements of the accelerometer under different resonance conditions, with a focus on the acceleration measurement method based on the mechanical frequency locking (MFL) phenomenon. First, different acceleration signals were simulated using variations in gravity, such as... Figure 7 As shown, the MEMS accelerometer, drive circuit, and detection circuit are integrated onto a PCB board via a ceramic housing. The integrated PCB board is placed together with the rotating electrode, and the angle of the rotating electrode is adjusted by a controller to achieve different acceleration signals. For example, when the tilt angle is 30 degrees, the gravitational acceleration is 0.5g, and the internal resonant beam is subjected to tension.

[0043] (1) Open-loop measurement: The sensitivity of the accelerometer was initially evaluated through open-loop measurements. A swept-frequency AC signal was applied to the driving electrodes of the mechanically coupled resonant beam, and the frequency response curve was experimentally measured. The change in peak frequency was recorded when different gravitational accelerations were applied to the mass block of the accelerometer.

[0044] First, consider DC voltage. The working principle of the accelerometer is given for a 15V case. Since the acceleration signal primarily affects the natural frequency of the internal resonant beam, the amplitude-frequency response curves of the internal resonant beam under different acceleration signals were experimentally measured, as shown below. Figure 8 As shown in (a) of the diagram, as the acceleration signal decreases from g to -g, the peak frequency of the resonant beam decreases from approximately 182.5 kHz to 182.2 kHz. Considering that the accelerometer is a symmetrical differential structure, its sensitivity is 312.6 Hz / g. Then, the DC voltage was experimentally measured. The effect on sensitivity. When DC voltage When the voltage is increased to 65V, the sensitivity of the resonant accelerometer can be improved to 366Hz / g, such as... Figure 8 As shown in (b) of the figure. To explain the principle of the improvement, the dimensionless sensitivity of the resonant accelerometer was theoretically obtained through the following equation.

[0045] ; ; in, , These represent the lateral displacements of the external and internal resonant beams, respectively. This represents the lateral acceleration of the internal resonant beam. This is the linear elastic stiffness term corresponding to the fourth spatial derivative of the internal resonant beam; The physical coupling stiffness between the resonant beams. For Dirac A function that represents coupling that acts only at a specific location. ; For geometric nonlinear coefficients, describe the additional stiffness effect caused by large deformation tension of the beam; External axial load (including residual stress); , These are the first-order (slope) and second-order (curvature) spatial derivatives of the displacement field, respectively. This is the electrostatic coefficient, which is related to geometric parameters such as dielectric constant and plate gap. For the first The normalized intrinsic mode function of the root beam, The square of the bare modal natural angular frequency of the beam when there is no electrostatic discharge and no coupling; Here is the axial coordinate of the internal resonant beam; , These are the natural angular frequencies of the bare modes of the external and internal resonant beams, respectively, when there is no electrostatic discharge and no coupling. , These are the equivalent cubic nonlinear stiffness coefficients of the external resonant beam and the internal resonant beam, respectively. This is a modal coupling cross term; This refers to the AC drive voltage applied to the external resonant beam.

[0046] Considering DC voltage The increase in [value] leads to a softening of the stiffness of the internal resonant beam. This is achieved through the following equation: [equation missing]. The decrease leads to an increase in sensitivity.

[0047] .

[0048] in, This is the frequency sensitivity coefficient; For acceleration; The equivalent mass of the internal resonant beam; The total length of the beam; The bending stiffness of the internal resonant beam; pass Figure 6 It can be seen that when the AC drive voltage is higher than 180mV, the accelerometer exhibits a significant mechanical frequency lock-in (MFL) phenomenon. It is important to note that when MFL occurs, the peak frequency of the external resonant beam is primarily determined by the fundamental frequency of the internal resonant beam. Therefore, changes in the acceleration signal can be detected by measuring the peak frequency of the external resonant beam. As the acceleration signal decreases from -g to g, the lock-in frequency drops from approximately 174.3kHz to 174.6kHz. Figure 9 As shown in (a) above. By comparison Figure 8 (b) and Figure 9 In (a), it was found that the mechanical frequency lock-in (MFL) phenomenon generally does not significantly affect the sensitivity of the accelerometer. Specifically, when the AC drive voltage is increased to 300mV, the peak frequency of the resonant system remains unchanged under the same acceleration signal, such as... Figure 9 As shown in (b) of the figure. By utilizing the mechanical frequency locking (MFL) phenomenon for acceleration signal detection, the dependence of peak frequency on driving force caused by nonlinear vibration can be overcome.

[0049] (2) Accelerometer closed-loop measurement: Open-loop measurements are typically used for preliminary feasibility studies of MEMS accelerometers. However, due to instrument complexity and long measurement times, they are not suitable for real-time acceleration signal measurement in practical applications. To further test the sensor's performance, we conducted closed-loop experiments on the accelerometer and achieved real-time monitoring of the acceleration signal. The robustness of the accelerometer was analyzed through these experiments.

[0050] First, consider DC voltage. For both 15V and 65V conditions, the experiment measured the real-time variation of the peak frequency of the MEMS accelerometer with the acceleration signal. Figure 10The real-time detection ladder diagram is shown when the acceleration signal changes in steps of 0.5g. The sensitivity of the closed-loop measurement is consistent with that of the open-loop measurement. However, due to the nonlinearity of the stiffness, the peak frequency of the accelerometer fluctuates significantly when the AC drive voltage increases from 5mV to 8mV, severely affecting the robustness of the accelerometer. Figure 10 As shown in (a) and (b) in the figure.

[0051] To improve the robustness of accelerometers caused by nonlinear vibrations and enhance their frequency stability, a closed-loop real-time detection method based on the mechanical frequency locking (MFL) phenomenon is proposed. As discovered in the previous section, the MFL phenomenon occurs when the AC voltage is 200mV, and the peak frequency of the external resonant beam is locked near the natural frequency of the internal resonant beam. Figure 10 As shown in (c), when the AC voltage increased to 300mV, the peak frequency of the accelerometer did not change significantly. This demonstrates that the mechanical frequency lock-in (MFL) phenomenon can greatly enhance the detection robustness of the accelerometer.

[0052] S4. Through open-loop and closed-loop tests in step S3, a preliminary study was conducted on the sensitivity and dynamic performance of the accelerometer. Resolution is an important indicator of an accelerometer, representing the smallest acceleration signal that the accelerometer can detect. The frequency stability of the resonant system is a significant factor affecting measurement resolution. This step aims to evaluate the minimum resolution of the accelerometer under different resonant conditions by testing the Allen deviation of the resonant frequency. It should be noted that the resolution calculated in this invention is based solely on the frequency stability of the device and does not consider circuit noise during the integration process.

[0053] The minimum resolution of the accelerometer under different resonance conditions is evaluated by testing the Allen deviation of the resonant frequency, including the following steps: S41. Perform static testing and measure the Allen deviation of the resonant frequency under different resonant conditions: Place the MEMS accelerometer horizontally and record the resonant frequency for 5 minutes. The sampling frequency is 1600 Sa / s.

[0054] Allen bias in MEMS accelerometer measurement data, such as Figure 11 As shown, the inset plot represents the real-time frequency of the accelerometer in the closed-loop measurement. DC voltage is considered. At 15V, the minimum Allen deviation is 289.13ppb within the integration time of 23 seconds, which is mainly affected by Gaussian noise and temperature drift. Similarly, considering DC voltage... At 65V and an AC voltage of 40mV, the minimum Allen deviation was 90.32ppb within an integration time of 37 seconds. Then, by driving an external resonant beam, when the AC driving voltage increased to 450mV, mechanical frequency lock-in (MFL) occurred, with a minimum Allen deviation of 5.69ppb. This demonstrates that the MFL phenomenon can suppress Allen deviation and enhance the frequency stability of the resonant system. This phenomenon can be explained as follows: without the MFL phenomenon, the response frequency depends on the vibration amplitude of the resonant beam; any disturbance in the amplitude will cause fluctuations in the response frequency. When the MFL phenomenon occurs, the dependence of the response frequency on the amplitude of the resonant system is suppressed, thus overcoming the adverse effects of amplitude disturbances on frequency fluctuations.

[0055] S42. Based on the proportional relationship between resolution and Allen deviation, the resolution of a single-sided resonant structure can be obtained. As shown in the following formula: ; in, and The Allen bias and sensitivity of the single-sided resonant structure are represented respectively. This is the inherent resonant frequency of the resonant structure. It is important to note that the symmetry of the resonant system must be considered. The sensitivity of a differential structure is twice that of a single-sided resonant structure. The Allen variance of a differential structure is equal to... The Allen variance of a single-sided resonant structure is multiplied by the Allen variance. Therefore, the minimum resolution of a differential resonant accelerometer can be predicted by measuring the sensitivity and Allen variance of the single-sided resonant structure, as shown in the following formula: .

[0056] S43. Substitute Allen bias and sensitivity into the formula to obtain the resolution under different resonance conditions.

[0057] pass Figure 10 and Figure 11 Substituting the Allen bias and sensitivity into the above two equations, we obtain the resolution under different resonance conditions, such as... Figure 12 As shown. Considering DC voltage. At 15V, the resolution of the MEMS accelerometer decreases with increasing AC drive voltage. When the AC drive voltage is 60mV, the minimum resolution of the MEMS accelerometer reaches 194.58µg. Similarly, considering DC voltage... At 65V, the resolution of the MEMS accelerometer decreases with increasing AC drive voltage, reaching a minimum resolution of 39.27µg. Then, DC voltage is considered. At 65V, when the driving voltage exceeds 200mV, the mechanical frequency lock-in (MFL) phenomenon occurs. By driving and detecting the external resonant beam, the minimum resolution of the accelerometer can be reduced to 3.7µg. Therefore, the experimental results show that MFL can significantly improve the resolution of the accelerometer. Table 2 compares the performance of the accelerometer under three different resonance conditions.

[0058] Table 2 Comparison of accelerometer performance under three different resonance conditions

[0059] The results show that increasing the DC voltage can significantly improve the sensitivity of the accelerometer. The detection method based on the mechanical frequency locking (MFL) phenomenon can enhance the robustness of the resonant accelerometer and improve its resolution.

[0060] In summary, this invention designs, manufactures, and experimentally characterizes a multi-electrode driven mechanically coupled differential MEMS resonant accelerometer comprising four resonant beams and an amplification mechanism. Through open-loop experimental measurements, the amplitude-frequency response characteristics of the accelerometer under different resonant conditions are discussed in detail. The stiffness hardening effect caused by geometric nonlinearity leads to the dependence of the response frequency on the vibration amplitude. In particular, through the 1:1 internal resonance behavior of the mechanically coupled resonator, the mechanical frequency locking (MFL) phenomenon of the resonant system is experimentally measured and theoretically predicted, overcoming the amplitude dependence of the response frequency.

[0061] In addition, DC voltage is considered Experimental results at 15V and 65V show that the sensitivities of the MEMS accelerometer are 312.6Hz / g and 366Hz / g, respectively. The electrostatic driving voltage can soften the resonant beam, thereby improving the sensitivity of the resonant accelerometer. Closed-loop experimental measurements revealed that when there is no mechanical frequency lock-in (MFL) phenomenon in the resonant system, disturbances in the driving force significantly affect the frequency stability of the resonant system. Conversely, when the MFL phenomenon occurs, the disturbances in the driving force have a very small impact on the response frequency. Based on this principle, a novel detection method utilizing frequency-locked motion is proposed.

[0062] Experimental results show that MFL can significantly improve the resolution of the accelerometer, with a minimum resolution reaching 3.7 µg. The findings of this invention demonstrate that the mechanical frequency locking (MFL) phenomenon is of great significance for improving the performance of nonlinear resonant sensors.

[0063] pass Figure 13 It can be seen that the dynamic model of the two resonant beams can be described by Euler-Bernoulli theory. Considering that the movable mass block and the internal resonant beam are connected by four force-amplifying levers, the internal resonant beam is subjected to tension or compression caused by axial inertial force. It is important to note that when an external resonant beam is driven by an AC voltage, mechanical frequency lock-in (MFL) occurs. The Hamiltonian principle is used to control the lateral deflection. and The equation of motion can be written as: (A1); (A2); in, Represents the unit step function. ( i =1,2) represents the position coordinates, A For cross-sectional area, I For rotational inertia, It is the dielectric constant; The normalized dimensionless coordinates of the resonant beam axis; For tensile stiffness; Let be the first spatial derivative of the external resonant beam's vibration displacement along the axial direction, representing the beam's bending slope; The second spatial derivative of the vibration displacement of the external resonant beam along the axial direction represents the beam's bending curvature. Let be the fourth-order spatial derivative of the vibration displacement of the external resonant beam along the axial direction, and be the standard term of the Euler-Bernoulli beam bending control equation. It should be noted that this invention only considers the linear coupling stiffness generated by the coupling beam between the two resonant beams, where... Let be the linear coupling stiffness coefficient. Considering that the lever mechanism can amplify the inertial force of the movable mass block by approximately 13 times and apply it to the internal resonant beam, the corresponding relationship can be derived: .

[0064] The boundary conditions are shown in the following equation: (A3); in, , , The first A resonant beam ( =1 represents the external resonant beam. =2 represents the internal resonant beam at the left end ( =0) and the right end ( = ) lateral displacement. , Let be the slopes of the resonant beam at the left and right ends; The first derivative of the lateral displacement of the resonant beam with respect to the axial coordinate. For time.

[0065] Then, dimensionless variables are introduced. , , .consider Much larger The following relation is obtained: .

[0066] Substituting the dimensionless variables into (A1)-(A3), and expanding the electrostatic force into third-order nonlinear terms, we obtain the following dimensionless equations of motion (A4)(A5): (A4); (A5); Boundary conditions (A6): (A6); in, , , Considering that both resonant beams are vibrating in the first bending mode, their deformation can be expressed as: ; in, Let be the shape of the first linear undamped mode of the straight beam. At this point, the linear undamped eigenvalue problem can be expressed as: .

[0067] Will Substitute into formulas (A4) and (A5), and multiply by And integrate ( =0 to 1), resulting in (A7)(A8): (A7); (A8); The coefficients for each item are shown in (A9): (A9); It should be noted that the coupling stiffness coefficient can be obtained through finite element simulation. Considering the relatively low DC bias voltage of the external resonant beam, the influence of static displacement and second-order nonlinear stiffness on the dynamic behavior can be neglected.

[0068] Therefore, this invention employs the aforementioned mechanically coupled MEMS resonant accelerometer and its detection method based on internal resonance. Compared with traditional resonant detection methods, the accelerometer based on the mechanical frequency locking (MFL) phenomenon exhibits better robustness and can suppress detection errors caused by AC drive voltage fluctuations. Simultaneously, the mechanical frequency locking (MFL) phenomenon improves the frequency stability of the accelerometer and reduces the Allen variance of the response frequency. Based on this phenomenon, the proposed high-resolution acceleration detection principle is of great significance to the development of nonlinear resonant sensors.

[0069] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A mechanically coupled MEMS resonant accelerometer based on internal resonance, characterized in that: It includes a movable mass block and force amplification levers set at the four corners of the movable mass block. Two pairs of mechanically coupled resonant beams are symmetrically arranged on both sides of the movable mass block. Several electrodes are arranged on both sides of each pair of mechanically coupled resonant beams. The mechanically coupled resonant beams include an external resonant beam and an internal resonant beam. The external resonant beam and the internal resonant beam are connected by the mechanically coupled beam.

2. The mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 1, characterized in that: The movable mass block is used to receive external acceleration signals and generate inertial force; The force amplification lever is used to amplify the inertial force generated by the movable mass block and transmit it to the internal resonant beam. The electrodes include a driving electrode, a detection electrode, and a stiffness adjustment electrode; the driving electrode is used to apply a swept-frequency AC driving signal; the detection electrode is used to acquire and output the vibration signal of the resonant beam; the stiffness adjustment electrode is used to apply a DC voltage to adjust the fundamental frequency of the internal resonant beam.

3. The mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 2, characterized in that: The driving electrode and the detection electrode are provided on both the left and right sides of the external resonant beam; An AC driving voltage is applied to the driving electrode. When the AC driving voltage exceeds 180mV, energy is transferred between the external resonant beam and the internal resonant beam through a 1:1 internal resonance, triggering the mechanical frequency lock (MFL) phenomenon.

4. The mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 3, characterized in that, The quality factor of the external resonant beam is 8000, and the quality factor of the internal resonant beam is 34000.

5. The mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 2, characterized in that: The stiffness adjustment electrodes are symmetrically distributed on the upper and lower sides of the internal resonant beam. A DC voltage in the range of 0V to 70V is applied to the stiffness adjustment electrodes to compensate for the fundamental frequency difference between the external resonant beam and the internal resonant beam caused by processing errors and residual stress, so that the external resonant beam and the internal resonant beam generate a 1:1 internal resonance and excite the mechanical frequency lock (MFL) phenomenon.

6. The mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 1, characterized in that: Both the movable mass block and the mechanically coupled resonant beam are grounded.

7. The detection method of a mechanically coupled MEMS resonant accelerometer based on internal resonance as described in any one of claims 1-6, characterized in that, Includes the following steps: S1. Apply a DC driving voltage to the internal resonant beam and adjust the fundamental frequency of the internal resonant beam so that the fundamental frequency of the external resonant beam is close to that of the internal resonant beam, so as to meet the triggering condition of 1:1 internal resonance. S2. Apply an AC drive signal to the drive electrode of the external resonant beam. When the amplitude of the AC drive signal is greater than or equal to the critical value of the mechanical frequency lock-in MFL phenomenon, the 1:1 internal resonance is excited to generate the mechanical frequency lock-in MFL phenomenon. S3. Apply different acceleration signals to the movable mass block, measure the peak frequency of the external resonant beam, and determine the acceleration signal to be measured based on the change in peak frequency. S4. By testing the Allen deviation of the resonant frequency, evaluate the minimum resolution of the accelerometer under different resonant conditions.

8. The detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 7, characterized in that, S1 includes the following steps: S11. Apply a DC voltage ranging from 0V to 70V between the symmetrical stiffness adjustment electrodes on the upper and lower sides of the internal resonant beam. ; A lock-in amplifier is used to generate a swept-frequency AC signal and apply it to the upper electrode; the lower electrode outputs a corresponding induced current signal, which is amplified by a cross-group amplifier and observed on the lock-in amplifier. S12. Apply a pair of DC voltages to the driving electrodes on the left and right sides of the external resonant beam. A swept-frequency AC signal is applied to the left electrode of the external resonant beam, and the corresponding induced current signal is output from the right electrode. After being amplified by the cross-group amplifier, it is observed on the lock-in amplifier.

9. The detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 8, characterized in that, In S2, the critical value of the mechanical frequency lock-in (MFL) phenomenon is the parameter value corresponding to the merging of the vibration peaks of the external and internal resonant beams in the amplitude-frequency response curve of the accelerometer, and the peak frequency locks at 174.5kHz. When the mechanical frequency lock-in (MFL) phenomenon occurs, the following dynamic equations of the mechanically coupled resonant system are satisfied: ； ； in, and These represent the amplitudes of the external and internal resonant beams, respectively. and These represent the instantaneous velocities of the external and internal resonant beams, respectively. and These represent the instantaneous accelerations of the external and internal resonant beams, respectively. and These represent the fundamental frequencies of the external and internal resonant beams, respectively. and These represent the damping of the external resonant beam and the internal resonant beam, respectively. This represents the mechanical coupling stiffness between the external and internal resonant beams. and Indicates the nonlinear stiffness coefficient; and This indicates the amplitude and frequency of the electrostatic driving force.

10. The detection method of the mechanically coupled MEMS resonant accelerometer based on internal resonance according to claim 9, characterized in that, S4 includes the following steps: S41. Perform static testing: Place the MEMS accelerometer horizontally and record the resonant frequency for 5 minutes. The sampling frequency is 1600 Sa / s. Measure the minimum Allen deviation under different resonant conditions. S42. Based on the proportional relationship between resolution and Allen deviation, combined with the resolution formula of a single-sided resonant structure, and considering the symmetry of the resonant system, the minimum resolution of the differential resonant accelerometer is predicted by measuring the sensitivity and Allen variance of the single-sided resonant structure. S43. Substitute Allen bias and sensitivity into the formula to obtain the resolution under different resonance conditions.

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